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Theoretical Computer Science

183 missions · 115 completed

The mathematical foundations of computation: which problems can be solved, by what algorithms, and at what cost in time, space, or communication. Distinguished by its emphasis on rigor and unconditional lower bounds, it spans computational complexity, algorithm design, automata and computability, cryptography, and the analysis of Boolean functions.

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Completed

Combinatorics Graph Theory Operations Research+1·Captain: mikedeng1

An Analysis of Several Heuristics for the Traveling Salesman Problem II: Every Insertion Method Is Within ⌈lg n⌉ + 1 of the Optimal TourResearch Paper

Motivation

The traveling salesman problem asks for a shortest closed route visiting every node of a weighted complete graph exactly once. It is NP-hard, so practitioners use fast heuristics, and the basic question about a heuristic is how far from optimal its tour can be. Rosenkrantz, Stearns and Lewis (SIAM J. Comput. 6(3), 1977) gave the first systematic worst-case analysis of the simple constructive heuristics under the triangle inequality: nearest neighbor, the family of insertion methods, and several variants.

Insertion methods build a tour by growing it one node at a time. They are among the most widely used construction heuristics in practice and in textbooks, and they differ only in the rule that chooses which node to insert next: the nearest one, the cheapest one, the farthest one, a random one, or any other. This mission formalizes the paper's result that holds for the whole family at once, regardless of that rule: every insertion method produces a tour at most $\lceil \lg n\rceil + 1$ times longer than an optimal one (Theorem 3, p. 571).

Timeline. 1977: Rosenkrantz, Stearns and Lewis prove $\lceil\lg n\rceil+1$ for every insertion method (Theorem 3), $\tfrac12(\lceil\lg n\rceil+1)$ for nearest neighbor (Theorem 1), both from a shared counting lemma (Lemma 1), and the constant $2$ for nearest and cheapest insertion (Theorem 4). 1994: Bafna, Kalyanasundaram and Pruhs (Theoretical Computer Science 125, 1994) give instances on which some insertion methods reach ratio $\Omega(\log n/\log\log n)$ , so the logarithmic growth cannot be replaced by a constant for the family as a whole.

Setting

A traveling salesman graph with $n$ nodes consists of a finite node set $N$ with $|N|=n$ and a distance $d:N\times N\to\mathbb R$ with $d(i,j)=d(j,i)$ , $d(i,j)\ge 0$ and $d(i,j)+d(j,k)\ge d(i,k)$ for all nodes (the triangle inequality). A tour visits every node once and returns to its start; its length is the sum of its edge lengths, and OPTIMAL is the least length of a tour.

A subtour is a tour on a subset of the nodes; a single node is a tour without edges. Given a subtour $T$ and a node $k\notin T$ , TOUR $(T,k)$ is obtained by choosing an edge $(x,y)$ of $T$ minimizing

d(x,k)+d(k,y)-d(x,y)

and replacing it by the edges $(x,k)$ and $(k,y)$ ; if $T$ is a single node $i$ , TOUR $(T,k)$ is the two-node tour $(i,k),(k,i)$ . COST $(T,k)$ is the length of TOUR $(T,k)$ minus the length of $T$ .

An insertion method constructs subtours $T_1,\dots,T_n$ with $T_1=\{a_0\}$ a single node and $T_{i+1}=\mathrm{TOUR}(T_i,a_i)$ for some node $a_i\notin T_i$ , $1\le i<n$ . The final tour $T_n$ is the approximation, and INSERT denotes its length. No rule for choosing the $a_i$ is fixed, and ties between minimizing edges are broken arbitrarily.

Write $\lg$ for the logarithm to base 2 and $\lceil x\rceil$ for the least integer $\ge x$ .

Formalization targets

Goal: Theorem 3

For every traveling salesman graph with $n\ge 1$ nodes and every run of every insertion method,

\mathrm{INSERT}\ \le\ \bigl(\lceil\lg n\rceil+1\bigr)\cdot\mathrm{OPTIMAL}.

Milestones

(2.2), shortcutting: visiting a subset of the nodes in the order of a tour gives a tour of the subset that is no longer.
(2.1): if the numbers $l_1\ge\dots\ge l_n$ satisfy $d(p,q)\ge\min(l_p,l_q)$ for distinct $p,q$ , then $\mathrm{OPTIMAL}\ge 2\sum_{i=k+1}^{\min(2k,n)} l_i$ for $1\le k\le n$ .
Lemma 1: if $d(p,q)\ge\min(l_p,l_q)$ for distinct nodes and $l_p\le\frac12\mathrm{OPTIMAL}$ for all $p$ , then

\sum_p l_p\le\tfrac12\bigl(\lceil\lg n\rceil+1\bigr)\mathrm{OPTIMAL}.

Lemma 2: $\mathrm{COST}(T,k)\le 2\,d(k,j)$ for every node $j$ of $T$ .
(3.7): $\mathrm{INSERT}=\sum_{i=1}^{n-1}\mathrm{COST}(T_i,a_i)$ .
(3.10): $\mathrm{COST}(T_i,a_i)\le 2\,d(a_i,a_j)$ whenever $j<i$ .
(3.12): $\mathrm{COST}(T_i,a_i)\le\mathrm{OPTIMAL}$ for $1\le i<n$ .

Significance

The result. Theorem 3 is a guarantee for an entire class of algorithms rather than for one. Any rule for choosing the next node, including rules designed for speed or for empirical quality, inherits a worst-case ratio of $\lceil\lg n\rceil+1$ from the insertion step alone. The rule matters only for improving on that: nearest and cheapest insertion achieve the constant $2(1-1/n)$ (Theorem 4 and its corollary, the subject of the third mission of this series), while the logarithmic bound remains the best general statement for other rules, such as farthest or arbitrary insertion. Lemma 1 is reusable on its own: it converts "every node carries a charge bounded by half the optimum and by its distance to other nodes" into a logarithmic bound, and the same lemma yields the nearest neighbor bound of Theorem 1.

Formalizing it. The theorem has been proved since 1977; the work here is a machine-checked proof of the known argument together with a reusable library for subtours, insertion and insertion costs. The companion nearest neighbor bound (Theorem 1) is already on the platform as SupplyChainTheory.nearest_neighbor_bound (proved), and nearest insertion with constant 2 as SupplyChainTheory.nearest_insertion_bound; neither covers arbitrary insertion methods or states Lemma 1 separately.

Difficulty

The per-step facts are local: each insertion is cheap relative to a node already present (Lemma 2) and relative to OPTIMAL (3.12). The obvious way to combine them, adding up $n-1$ costs each at most OPTIMAL, gives only the ratio $n-1$ . The logarithm comes from a global counting argument over all nodes simultaneously (Lemma 1), in which OPTIMAL is compared with tours on nested subsets of nodes of doubling size, and the per-node charges must be matched against the edges of those tours. Formally, the delicate parts are the bookkeeping of subtours as they grow (that every earlier node lies on the current subtour, and that the insertion cost equals the length increase), the shortcutting of a tour to an arbitrary subset, and the ceiling-of-logarithm arithmetic.

Formalization scope

Nodes are Fin n; a tour of all nodes is a permutation τ : Equiv.Perm (Fin n), and OPTIMAL is the minimum of the tour length over the finite, nonempty set of permutations. Subtours are duplicate-free lists of nodes, with closed length $d(x_0,x_1)+\dots+d(x_{m-1},x_0)$ . TOUR $(T,k)$ is encoded as inserting $k$ at a list position whose resulting length is minimal among all positions; inserting at a position removes exactly one edge of $T$ and raises the length by exactly $d(x,k)+d(k,y)-d(x,y)$ , so this is the paper's rule, with every tie-breaking allowed. COST is the minimum length increase over positions. The paper's 1-based subtour index is kept ( $T_1=[a_0]$ , $T_n$ final). $\lceil\lg n\rceil$ is Nat.clog 2 n. All quantities are real.

Conventions and deviations, each disclosed in the item statements:

The distance satisfies $d(i,i)=0$ , a normalization not in the paper; a loop never enters any length.
Ratios are multiplied out ( $\mathrm{INSERT}\le c\cdot\mathrm{OPTIMAL}$ ), so the paper's exclusion of the identically zero distance (1.1) is not needed.
Condition a) of Lemma 1 is required for distinct nodes only. The page says "for all nodes $p$ and $q$ ", which for $p=q$ would force every $l_p\le 0$ and make the lemma inapplicable in the proof of Theorem 3; the proof uses the condition only on edges of a tour.
(2.2) is stated for every subset of the nodes and every tour, which is what the shortcut argument shows; the paper applies it to one specific subset and an optimal tour.
(2.1) uses 0-based node labels, so its range $k+1,\dots,\min(2k,n)$ becomes $k,\dots,\min(2k,n)-1$ .

The goal quantifies over every run: any choice of the inserted nodes $a_i$ and any minimizing insertion position. Adding a selection rule (nearest, cheapest) or fixing a tie-breaking would state a weaker, different theorem; restricting to instances with OPTIMAL $=0$ or to a fixed small $n$ would trivialize it.

Reusable beyond this mission: the subtour and insertion library (closed length of a list, TOUR, COST, insertion runs) and Lemma 1, which also yields Theorem 1. Contributions welcome: proofs of the milestones, general lemmas about the closed length of List.insertIdx and of filtered lists, and a proof of Theorem 1 from this mission's Lemma 1.

Selected references

D. J. Rosenkrantz, R. E. Stearns, P. M. Lewis II, An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM Journal on Computing 6(3):563–581, 1977. https://doi.org/10.1137/0206041
V. Bafna, B. Kalyanasundaram, K. Pruhs, Not all insertion methods yield constant approximate tours in the Euclidean plane, Theoretical Computer Science 125(2):345–353, 1994.

10 thms2 active usersReviewed

Completed

Combinatorics Dynamic Programming·Captain: mikedeng1

A Faster Algorithm Computing String Edit Distances 1: for a finite alphabet and discrete costs, the block algorithm (Algorithms Y and Z) computes the edit distance from a finite tableResearch Paper

Motivation

The edit distance between two strings is the least total cost of a sequence of single-character insertions, deletions and replacements turning one string into the other. It is the basic similarity measure of spelling correction, file comparison and biological sequence alignment. Wagner and Fischer (JACM 1974) showed that it can be computed by filling a $(|A|+1) \times (|B|+1)$ matrix in $O(|A|\cdot|B|)$ time. Masek and Paterson (J. Comput. System Sci. 1980) gave the first asymptotic improvement: for a finite alphabet and edit costs that are integral multiples of a common constant, the edit distance can be computed in time $O(|A|\cdot|B|/\max(1, |B|/\log|A|))$ , that is $O(n^2/\log n)$ for two strings of length $n$ .

Timeline:

1970: Arlazarov, Dinic, Kronrod and Faradzev compute transitive closures by precomputing all small submatrices, the "four Russians" technique that Masek and Paterson adapt.
1974: Wagner and Fischer give the matrix-filling algorithm, with the first row and column of the matrix and the three-term recurrence for its interior (Theorems 1 and 2 of Masek–Paterson, cited from them).
1980: Masek and Paterson apply the four-Russians technique to the edit matrix, working with differences of adjacent entries, and show that the restriction to discrete costs cannot simply be dropped (their Section 4, the subject of the second mission of this series).
2015: Backurs and Indyk (arXiv:1412.0348) show that a strongly subquadratic algorithm would refute the Strong Exponential Time Hypothesis, so a logarithmic-factor speed-up of this kind is close to the best one can expect.

Setting

Let $\Sigma$ be an alphabet and $\lambda$ the null string. For a string $A$ , $|A|$ is its length, $A_n$ its $n$ -th character, $A^{i,j} = A_i \cdots A_j$ and $A^i = A^{1,i}$ , with $A^0 = \lambda$ .

An edit operation $a \to b$ is a pair $(a, b) \ne (\lambda, \lambda)$ of strings of length at most one: a replacement ( $a, b \ne \lambda$ , possibly $a = b$ ), a deletion ( $b = \lambda$ ) or an insertion ( $a = \lambda$ ). $B$ results from $A$ via $a \to b$ if $A = \sigma a \tau$ and $B = \sigma b \tau$ . An edit sequence $S = s_1, \dots, s_m$ takes $A$ to $B$ if there are strings $A = C_0, C_1, \dots, C_m = B$ with $C_{i-1} \to C_i$ via $s_i$ . A cost function $\gamma$ assigns a nonnegative real to every edit operation, $\gamma(S) = \sum_i \gamma(s_i)$ , and

\delta(\gamma, A, B) = \min\{\gamma(S) \mid S \text{ takes } A \text{ to } B\}.

Write $R_{a,b} = \gamma(a \to b)$ , $D_a = \gamma(a \to \lambda)$ , $I_a = \gamma(\lambda \to a)$ , and $\delta_{i,j} = \delta(\gamma, A^i, B^j)$ for the entries of the edit matrix. The cost function is normalized if $\gamma(a \to b) = \delta(\gamma, a, b)$ for every edit operation.

A step is a difference of two adjacent matrix entries, $\delta_{i,j} - \delta_{i-1,j}$ (vertical) or $\delta_{i,j} - \delta_{i,j-1}$ (horizontal). The cost set is $\Omega = \{D_a\} \cup \{I_a\} \cup \{R_{a,b}\}$ , and $\Omega$ is discrete if every element of $\Omega$ is an integral multiple of one constant $r > 0$ .

Algorithm Y takes two strings $C, D$ of length $m$ and two step vectors $R, S$ of length $m$ (the left column and top row of an $m \times m$ block) and fills a matrix $T$ of vertical steps and $U$ of horizontal steps by the recurrence of Corollary 1, returning the right column $R'$ and bottom row $S'$ . Algorithm Z cuts $A$ and $B$ into blocks of length $m$ , starts from the deletion costs of $A$ and the insertion costs of $B$ , obtains the steps of each block from Algorithm Y's output ("Fetch"), and returns the sum of the steps along the left column and the bottom row.

Formalization targets

Goal: correctness from a string-independent finite table

For a finite alphabet and a nonnegative, normalized cost function with discrete $\Omega$ , there is a finite set $T \subset \mathbb{R}$ such that for all $m \ge 1$ and all $A, B$ with $m \mid |A|$ , $m \mid |B|$ ,

\mathrm{cost}_{Z}(\gamma, m, A, B) = \delta(\gamma, A, B), \qquad \text{every entry of every } P(i,j), Q(i,j) \text{ lies in } T.

$T$ is fixed before $m$ , $A$ and $B$ . The second clause says that Algorithm Y's table need only range over $\Sigma^m \times \Sigma^m \times T^m \times T^m$ , whose size does not depend on the strings; this is what the running-time bound rests on.

Milestones

Theorem 1 [Wagner–Fischer]: $\delta_{0,0} = 0$ , $\delta_{i,0} = \sum_{r \le i} D_{A_r}$ , $\delta_{0,j} = \sum_{r \le j} I_{B_r}$ .
Theorem 2 [Wagner–Fischer]: $\delta_{i,j} = \min(\delta_{i-1,j-1} + R_{A_i,B_j}, \delta_{i-1,j} + D_{A_i}, \delta_{i,j-1} + I_{B_j})$ .
Corollary 1: the same recurrence written in terms of steps.
Algorithm Y returns the final step vectors of every $m \times m$ submatrix from its initial step vectors and strings (Section 2.1).
Lemma 3: $-I \le \delta_{i,j} - \delta_{i-1,j} \le D$ and $-D \le \delta_{i,j} - \delta_{i,j-1} \le I$ .
Lemma 4: if $\Omega$ is discrete, the set of possible steps is finite.

Significance

The result was the first algorithm for edit distance faster than quadratic, and its method (tabulate every possible small block of a dynamic program, described by differences rather than values) became the standard way of shaving a logarithmic factor from string dynamic programs. The discreteness hypothesis is where the method's power ends: Section 4 of the paper shows that with costs $1$ and $\pi$ the number of distinct steps grows without bound.

The results are proved in the paper; none of them is formalized. The Mathlib revision of this mission contains no edit-distance module. A formalization produces a definition of edit distance as a minimum over edit sequences, a machine-checked proof of the Wagner–Fischer recurrence for that definition under the normalization the paper assumes, and a checked proof that the four-Russians block assembly is correct and draws on a finite, string-independent table.

Difficulty

The hardest step is Theorem 2 for $\delta$ defined as a minimum over arbitrary edit sequences. An edit sequence may insert a character and later replace or delete it, or edit the same position many times, so the edit matrix's three-term recurrence does not follow by looking at the last operation. The upper bound is direct; the lower bound needs a normal form for edit sequences, and it fails without normalization: with $R_{a,c} = 10$ , $R_{a,b} = R_{b,c} = 1$ and all insertions and deletions costing $100$ , $\delta(\gamma, a, c) = 2$ while the recurrence gives $10$ .

The goal is then an induction over blocks that must keep track of which matrix entries each block's input and output vectors represent, with block boundaries at multiples of $m$ and the first row and column handled by Theorem 1.

Formalization scope

Strings are List α; characters are 1-based in all statements, as in the paper ( $A_i$ is A[i-1]). An edit operation is a structure with two Option α fields, not both none. $\delta$ is sInf of the set of costs of edit sequences taking $A$ to $B$ (nonempty, and bounded below for $\gamma \ge 0$ ). Costs are real-valued with an explicit nonnegativity hypothesis. Normalization is a hypothesis on Theorems 1, 2, Corollary 1, the Algorithm Y lemma and the goal; Lemmas 3 and 4 hold without it. The finite alphabet is [Fintype α] on Lemma 4 and the goal; Lemma 3 is stated for arbitrary upper bounds $I \ge I_a$ , $D \ge D_a$ , which implies the paper's version with maxima. The paper's standing assumption $|A| \ge |B|$ serves only the running time and is dropped. Step vectors are functions on Fin m.

Pinned statements. The paper states a running time $O(|A|\cdot|B|/\max(1,|B|/\log|A|))$ on a logarithmic-cost RAM; the goal formalizes the two facts that bound rests on, correctness and a finite table domain fixed before the strings. The RAM model, operation counts, the choice $m = \lfloor \log_k |A| \rfloor$ , the padding reduction for $m \nmid |A|$ , and the edit-path recovery of Section 2.3 are not formalized. Algorithm Y's result is a function (blockY); the Store/Fetch memory is not modelled.

The edit distance must stay the minimum over edit sequences: defining it by the Wagner–Fischer recurrence would make Theorems 1 and 2 true by definition and reduce the goal to a comparison of two recurrences. Algorithms Y and Z are transcribed from the pseudo-code and never refer to $\delta$ .

Useful beyond this mission: the §1.1 definitions and Theorems 1–2 are a general edit-distance library (the second mission of this series defines the same objects). Contributions of lemmas about normal forms of edit sequences, the triangle inequality for $\delta$ , and attainment of the minimum are welcome.

Selected references

W. J. Masek, M. S. Paterson, A Faster Algorithm Computing String Edit Distances, J. Comput. System Sci. 20 (1980), 18–31. https://doi.org/10.1016/0022-0000(80)90002-1
R. A. Wagner, M. J. Fischer, The String-to-String Correction Problem, J. ACM 21 (1974), 168–173. https://doi.org/10.1145/321796.321811
V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, I. A. Faradzev, On Economical Construction of the Transitive Closure of an Oriented Graph, Soviet Math. Dokl. 11 (1970), 1209–1210.
A. Backurs, P. Indyk, Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false), STOC 2015. https://arxiv.org/abs/1412.0348

10 thms2 active usersReviewed

Combinatorics Graph Theory Probability·Captain: mikedeng1

A Simple Parallel Algorithm for the Maximal Independent Set Problem II: The Round Bound of the Derandomized AlgorithmResearch Paper

Motivation

A maximal independent set (MIS) of a graph is a set of pairwise non-adjacent vertices to which no further vertex can be added. Sequentially an MIS is found greedily in linear time, but the greedy scan is inherently serial. Whether an MIS can be computed by a fast parallel algorithm was a central question of parallel complexity in the early 1980s: Karp and Wigderson gave the first NC algorithm (STOC 1984), and Luby's paper, SIAM J. Comput. 15(4):1036–1053, 1986, gave a much simpler one. MIS is a subroutine of many parallel and distributed graph algorithms (colouring, matching, symmetry breaking), and Luby's randomized algorithm remains the standard one in distributed computing.

The paper's second contribution, the subject of this mission, is a general method for removing randomness: analyse the randomized algorithm under pairwise independence only, then realize pairwise independent random variables on a sample space of polynomial size and try every sample point in parallel. The same method, often attributed jointly to Luby (1986) and to Alon, Babai and Itai (J. Algorithms 7, 1986), became a standard tool of derandomization.

Setting

Let $G = (V, E)$ be a finite simple graph with $n = |V|$ vertices labelled $0, \dots, n-1$ . The algorithm keeps a set $I$ (initially empty) and the current graph $G' = (V', E')$ , the subgraph of $G$ induced on $V'$ (initially $V' = V$ ). For $W \subseteq V'$ the neighbourhood is $N(W) = \{ i \in V' : \exists j \in W, (i,j) \in E' \}$ . Each execution of the loop body selects an independent set $I' \subseteq V'$ , adds it to $I$ , and deletes $I' \cup N(I')$ from $V'$ ; the loop runs while $V' \ne \emptyset$ . Write $d(i)$ for the degree of $i$ in $G'$ , $Y_k$ for the number of edges of $G'$ before the $k$ -th execution, and $\mathrm{sum}(i) = \sum_{j \in \mathrm{adj}(i)} 1/d(j)$ .

Algorithm B's select step draws a coin $\mathrm{coin}(i) \in \{0,1\}$ for each vertex, with $\Pr[\mathrm{coin}(i) = 1] = 1/2d(i)$ , puts $X = \{ i : \mathrm{coin}(i) = 1 \}$ , and removes from $X$ the endpoint of smaller degree of every edge inside $X$ (both endpoints on a tie).

The sample space. Fix a prime $q$ with $n \le q \le 2n$ . The sample points are the pairs $(x, y)$ with $0 \le x, y \le q-1$ , each of probability $1/q^2$ . With $n(i) = \lfloor q/2d(i) \rfloor$ , the coin of vertex $i$ at $(x,y)$ is $1$ iff $(x + y \cdot i) \bmod q < n(i)$ , so $\Pr[\mathrm{coin}(i) = 1] = p'_i = \lfloor q/2d(i) \rfloor / q$ , and distinct coins are pairwise independent.

Algorithm D. Each execution of the loop body first moves the isolated vertices of $G'$ into $I$ . Then:

Case 1. If a vertex $i$ of maximum degree has $d(i) \ge n/16$ , it joins $I$ , and $\{i\} \cup N(\{i\})$ is deleted.
Case 2. Otherwise all $q^2$ sample points are tried, the one whose coins make Algorithm B's select step eliminate the most edges is kept, and its $I'$ is used.

No random bits are used.

Formalization targets

Goal: the round bound and correctness of Algorithm D

For every graph $G$ on $n$ vertices, every prime $q$ with $n \le q \le 2n$ , and every run of Algorithm D (every tie-break among maximum-degree vertices and every maximizing sample point), the loop body is executed exactly $k$ times, with

k \ \le\ \frac{\log(n^2)}{\log(18/17)} + 16 \ \le\ 25 \cdot \log_2 n + 16,

and the output $I$ is a maximal independent set of $G$ .

Milestones

The sample space: Lemma 1, $\Pr[X_i = R_j] = n_{ij}/q$ , and Lemma 2, $\Pr[X_i = R_j, X_{i'} = R_{j'}] = n_{ij} n_{i'j'}/q^2$ for $i \ne i'$ .
The Technical Lemma: for $p_1 \ge \dots \ge p_n \ge 0$ and $c > 0$ , $\max_l (\alpha_l - c\beta_l) \ge \tfrac12 \min\{\alpha_n, 1/c\}$ .
The two steps of the proof of Theorem 1: $E[Y_k - Y_{k+1}] \ge \tfrac12 \sum_i d(i) \Pr[i \in N(I')]$ , and $\tfrac12 \sum_{\mathrm{sum}(i) \le 2} d(i)\,\mathrm{sum}(i) + \sum_{\mathrm{sum}(i) > 2} d(i) \ge |E'|$ .
Lemma C and Theorem 2: with pairwise independent coins of law $1/2d(i)$ ,

\Pr[i \in N(I')] \ge \tfrac18 \min\{\mathrm{sum}(i), 1\}, \qquad E[Y_k - Y_{k+1}] \ge \tfrac{1}{16} Y_k .

The rounding bound $\tfrac89 p_i \le p'_i \le p_i$ when $d(i) < n/16$ .
Lemma D and Theorem 3: with pairwise independent coins of law $p'_i$ and all $d(i) < n/16$ ,

\Pr[i \in N(I')] \ge \tfrac19 \min\{\mathrm{sum}(i), 1\}, \qquad E[Y_k - Y_{k+1}] \ge \tfrac{1}{18} Y_k .

In Case 2 some sample point eliminates at least $1/18$ of the edges; Case 1 occurs at most 16 times in any run before it terminates.

Significance

The goal is the deterministic half of Luby's result: an MIS is computed in $O(\log n)$ parallel rounds with no randomness, which places MIS in deterministic NC. The pairwise-independent analysis (Lemmas C, D, Theorems 2, 3) is the reusable part: it shows that the Monte Carlo algorithm's progress guarantee survives when mutual independence is weakened to pairwise independence, which is what makes a sample space of size $q^2 = O(n^2)$ sufficient. Lemmas 1 and 2 are the standard construction of pairwise independent variables with prescribed rational marginals.

All of these results are proved in the paper. None is formalized on the platform. A related but different object is the platform's dot-product hash family (AlmostLossless.pairwiseIndependent_dotHash), which has uniform marginals over a field and is not the $q^2$ -point matrix space with prescribed marginals $n_{ij}/q$ . The companion mission A Simple Parallel Algorithm for the Maximal Independent Set Problem I formalizes Theorem 1, the mutually independent analysis of Algorithms A and B.

Difficulty

The obvious route to Theorem 2 repeats the proof of Lemma B, which lower-bounds $\Pr[i \in N(I')]$ by a product over independent events. Under pairwise independence the probability of an intersection of three or more coin events is not determined by the marginals, so that product argument fails, and the constant degrades from $\tfrac18$ to $\tfrac1{16}$ .

The round bound needs a separate argument for high-degree vertices. The rounded probabilities $p'_i$ are close to $p_i$ only when $q/2d(i)$ is large, which is why vertices of degree at least $n/16$ are handled by Case 1. Counting the Case 1 rounds uses the vertex count $n$ of the original graph, not of the current one. Correctness at termination requires an invariant linking $I$ , $V'$ and $G$ across both kinds of rounds and the deletion of isolated vertices.

Formalization scope

Vertices are Fin n with labels $0, \dots, n-1$ , which is §4.2's indexing of $X_0, \dots, X_{n-1}$ ; the label enters $\mathbb{Z}/q\mathbb{Z}$ as a residue, and labels are distinct mod $q$ because $n \le q$ . The current graph is the induced subgraph kept on the full vertex type, with deleted vertices isolated. One execution of the loop body is a relation between states $(I, V')$ that leaves the maximizing vertex (Case 1) and the maximizing sample point (Case 2) free, as the page does, and a run is any sequence of states starting at $(\emptyset, V)$ that follows the relation while $V' \ne \emptyset$ . The goal asks for the first index $k$ with $V' = \emptyset$ , so a statement about a later state or a bound on $k$ without termination does not meet it.

The conditions $d(i) \ge n/16$ and $d(i) < n/16$ are encoded exactly as $n \le 16\,d(i)$ and $16\,d(i) < n$ in $\mathbb{N}$ . $\lfloor q/2d(i) \rfloor$ is natural-number division. The printed code tests $(x + y\cdot i) \bmod q \le n(i)$ , which puts $n(i) + 1$ residues in $X$ and contradicts $p'_i = \lfloor p_i q \rfloor / q$ stated on the same page; the formalization uses the strict test.

Lemmas C, D and Theorems 2, 3 quantify over every probability space carrying measurable, pairwise independent (IndepFun for each pair of distinct vertices) coins with the stated marginals at vertices of positive degree. Replacing pairwise by mutual independence, or fixing the probability space, would weaken them. They are stated for a fixed current graph, that is, as the expectation conditional on the state before the round, which is what their proofs establish. Expectations are Bochner integrals of a function with finitely many values and are therefore genuine. Lemma 2 carries the hypothesis $i \ne i'$ , implicit on the page.

The development needs the induced subgraph and degree bookkeeping from Mathlib's SimpleGraph, pairwise independence from ProbabilityTheory.IndepFun, finite counting in ZMod q, and real logarithms. The pairwise-independent analysis (Lemma C to Theorem 3) and the sample-space lemmas are reusable beyond this mission. Contributions to any milestone are welcome.

Selected references

M. Luby, A Simple Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. Comput. 15(4):1036–1053, 1986. https://doi.org/10.1137/0215074
R. M. Karp and A. Wigderson, A Fast Parallel Algorithm for the Maximal Independent Set Problem, J. ACM 32(4):762–773, 1985. https://doi.org/10.1145/4221.4226
N. Alon, L. Babai and A. Itai, A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem, J. Algorithms 7(4):567–583, 1986. https://doi.org/10.1016/0196-6774(86)90019-2

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms 3: First-Fit Decreasing and Best-Fit Decreasing Use at Most 11/9 L* + 4 BinsResearch Paper

Motivation

Bin packing asks for the fewest unit-capacity bins that hold a given list of item sizes. It models table formatting, the placement of program segments on pages, and the allocation of files to disc tracks, and it is NP-complete, so exact solutions require search in general. Johnson, Demers, Ullman, Garey and Graham (SIAM J. Comput. 3 (1974)) therefore studied four simple placement heuristics and bounded how far each can be from the optimum in the worst case. Their paper is one of the founding results of the worst-case analysis of approximation algorithms.

This mission concerns the two decreasing heuristics, which sort the items from largest to smallest before placing them. For them the paper proves that at most $\tfrac{11}{9}$ of the optimum, plus an additive constant, is ever used, and that the factor $\tfrac{11}{9}$ cannot be improved.

Timeline.

1973: D. S. Johnson's MIT thesis proves $FFD(L)\le \tfrac{11}{9}L^*+4$ ; the argument exceeds 75 pages.
1974: Johnson, Demers, Ullman, Garey and Graham publish the bound for FFD and BFD, with a complete proof of the reduction from BFD to FFD and an outline of the FFD argument.
1985: B. S. Baker gives a shorter proof of $FFD(L)\le\tfrac{11}{9}L^*+3$ (J. Algorithms 6).
1991: M. Yue publishes a proof of $FFD(L)\le\tfrac{11}{9}L^*+1$ .
2007: G. Dósa determines the tight additive constant, $FFD(L)\le\tfrac{11}{9}L^*+\tfrac{6}{9}$ (ESCAPE 2007, LNCS 4614).

Setting

A list is a finite sequence $L=(a_1,a_2,\dots,a_n)$ of real numbers in $(0,1]$ ; values may repeat. A bin has capacity $1$ ; its level is the sum of the numbers placed in it. The optimum $L^*$ is the least number of bins into which the elements of $L$ can be distributed so that no bin has level exceeding $1$ .

The bins $B_1,B_2,\dots$ start empty and the elements are placed one at a time, in list order.

First-Fit (FF) places $a_i$ into the bin $B_j$ of least index whose level $\beta$ satisfies $\beta\le 1-a_i$ .
Best-Fit (BF) places $a_i$ into a bin whose level $\beta$ satisfies $\beta\le 1-a_i$ and is as large as possible, the one of least index among ties.
First-Fit Decreasing (FFD) and Best-Fit Decreasing (BFD) first arrange $L$ into nonincreasing order and then apply FF, respectively BF.

$FFD(L)$ and $BFD(L)$ are the numbers of bins that receive at least one element.

Two auxiliary notions from the paper's proof also appear among the milestones. The position $(j,k)$ of an element in a packing means that it is the $k$ -th element placed into bin $j$ . The weight $W(X)$ of a collection of elements is defined through $k$ -pieces, the elements in $(\tfrac1{k+1},\tfrac1k]$ . Each element has the weight $w_1(x)=\lfloor 1/x\rfloor^{-1}$ . A pair $(x,y)$ with $x$ a $k$ -piece and $kx+y\le1$ has the discounted weight $w_2(x,y)=w_1(x)+\tfrac{k-1}{k}w_1(y)$ , and any other pair has $w_1(x)+w_1(y)$ . $W(X)$ is the least total weight over all ways of grouping $X$ into singletons and pairs.

Formalization targets

Goal: Theorem 3.2

For every list $L$ ,

FFD(L)\le \frac{11}{9}L^*+4\qquad\text{and}\qquad BFD(L)\le\frac{11}{9}L^*+4 .

The constants are the paper's. Both halves are part of the goal.

Milestones, in the order the argument uses them

Lemma 3.3. If $FFD(L)>rL^*+d$ with $r,d\ge1$ , the list $L'$ keeping only the elements exceeding $(r-1)/r$ also has $FFD(L')>rL'^*+d$ ; the same for BFD. With $r=\tfrac{11}{9}$ this reduces the goal to lists in $(\tfrac2{11},1]$ .
Claims 3.4.5 and 3.4.6, two steps of the proof of Theorem 3.4 that concern only the FFD packing $PF$ and the BFD run. On $[\tfrac16,1]$ , BFD places every element exceeding $\tfrac13$ exactly where FFD does. Among the remaining positions of $PF$ , the lexicographic order of positions respects the order of the sorted list.
Theorem 3.4. If $L\subseteq[\tfrac16,1]$ , then $BFD(L)\le FFD(L)$ . This transfers the bound from FFD to BFD on $(\tfrac2{11},1]$ .
Lemma 4.2. For every integer $N\ge4$ and $L\subseteq(\tfrac1N,\tfrac12]$ ,

W(L)\ge FFD(L)-N+2 .

The reduced assertion (Section 4, p. 314). If $L\subseteq(\tfrac2{11},1]$ , then

FFD(L)\le\frac{11}{9}L^*+4 .

Theorem 3.1, the matching lower bound: for each $k\ge1$ there is a list with $L^*=k$ and $FFD(L)=BFD(L)>\tfrac{11}{9}L^*-2$ .

Significance

The bound makes FFD and BFD, which run in $O(n\log n)$ time, the reference heuristics for off-line bin packing. The $\tfrac{11}{9}$ bound and its proof technique of weighting functions were the model for the analysis of many later packing and scheduling heuristics. Theorem 3.1 shows that the factor is exact, so together with the goal it determines $\lim_{k\to\infty}R_{FFD}(k)=\lim_{k\to\infty}R_{BFD}(k)=\tfrac{11}{9}$ , where $R_A(k)$ is the largest ratio $A(L)/L^*$ over lists with $L^*=k$ .

The result is proved, but the source proves it only in part. The paper gives complete proofs of Lemma 3.3, Theorem 3.4 and Theorem 3.1. For the reduced assertion it gives only an outline, whose central inequalities involve maps the paper never defines, and it refers to the thesis for the details. Lemma 4.2 is proved in the paper through two claims. A formal proof of the goal must therefore either formalize one of the later complete proofs (Baker 1985, Yue 1991, Dósa 2007) or reconstruct the thesis argument. No machine-checked proof of the $\tfrac{11}{9}$ bound is present in Mathlib or on the platform.

Difficulty

The obvious approach, used for First-Fit in Section 2 of the same paper, assigns each element a weight depending only on its size, so that every bin of the algorithm's packing weighs at least $1$ and every bin of an optimal packing weighs at most the target ratio. For FFD no weighting of single elements works at ratio $\tfrac{11}{9}$ . Summing $w_1$ over the elements overcharges the FFD packing: a set of elements fitting into one bin can carry total $w_1$ -weight well above $\tfrac{11}{9}$ . The paper's remedy is a weight defined on pairs, $W(X)$ , which discounts elements that could share a bin with a larger one. Even with $W$ , the bins of FFD whose largest element exceeds $\tfrac12$ do not fit the scheme. Handling them requires a case analysis that the paper only sketches and that runs to more than 75 pages in the thesis.

The BFD half cannot be obtained by bounding BFD by FFD in general: there are lists with $BFD(L)=\tfrac{10}{9}FFD(L)$ . Theorem 3.4 works only because Lemma 3.3 first removes all elements below $\tfrac2{11}$ .

Formalization scope

Lists are L : List ℝ with the predicate IsList L ( $0<a\le1$ for every element), assumed by every statement. $L^*$ is optBins L, the least b : ℕ admitting a map from the items to Fin b with every bin sum at most $1$ . A run keeps the nonempty bins as a List (List ℝ) in index order and opens a new bin at the end exactly when no nonempty bin fits, which matches the paper's "least $j$ " over infinitely many empty bins. The fit test is non-strict. FFD and BFD are FF and BF applied to sortDesc L, a stable merge sort into nonincreasing order. They are defined for every list, so the goal is stated for arbitrary, unsorted $L$ . Positions are $0$ -based pairs (bin, place in bin) read off the run.

$W$ sorts its argument into nonincreasing order, so index is the position in that order. It then minimizes over involutions of the positions, which encode the partitions into one- and two-element sets. Weights are real-valued; the paper's use of rationals is incidental. The range hypotheses are exactly the paper's: $[\tfrac16,1]$ is closed in Theorem 3.4, $(\tfrac2{11},1]$ is open at $\tfrac2{11}$ , and Lemma 4.2 has $\tfrac1N<a\le\tfrac12$ .

A weakened goal, such as $FFD(L)\le\tfrac{11}{9}L^*+c$ with a larger $c$ , a bound for sorted lists only, or the FFD half alone, is a different theorem and does not close the mission. Claims 3.4.1–3.4.4 and 3.4.7 and the inequalities $(*)$ , $(**)$ of the outline are not stated: they concern the paper's step-by-step construction and the undefined maps $f$ , $g$ .

A complete development needs basic lemmas about FF and BF runs (levels stay at most $1$ , a new bin opens only when nothing fits, runs on prefixes). It also needs invariance of FFD and BFD under permutations of equal elements, the monotonicity of $L^*$ under deletion, and $L^*\ge\sum_i a_i$ . These are reusable in the other missions of this series. Proofs of individual milestones, alternative complete proofs of the goal, and sharper additive constants are all welcome.

Selected references

D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, R. L. Graham, Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms, SIAM Journal on Computing 3(4):299–325, 1974. https://doi.org/10.1137/0203025
D. S. Johnson, Near-Optimal Bin Packing Algorithms, Ph.D. thesis, Massachusetts Institute of Technology, 1973 (reference [8] of the paper above).
B. S. Baker, A new proof for the first-fit decreasing bin-packing algorithm, Journal of Algorithms 6(1):49–70, 1985. https://doi.org/10.1016/0196-6774(85)90018-5
M. Yue, A simple proof of the inequality FFD(L) ≤ 11/9 OPT(L) + 1, ∀L, for the FFD bin-packing algorithm, Acta Mathematicae Applicatae Sinica 7(4):321–331, 1991.
G. Dósa, The tight bound of first fit decreasing bin-packing algorithm is FFD(I) ≤ 11/9 OPT(I) + 6/9, ESCAPE 2007, LNCS 4614:1–11, 2007. https://doi.org/10.1007/978-3-540-74450-4_1

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms 1: First-Fit and Best-Fit Have Asymptotic Worst-Case Ratio 17/10Research Paper

Motivation

Bin packing asks for the fewest unit-capacity bins that hold a given list of item sizes. It is one of the first problems studied through the worst-case analysis of approximation algorithms, and it models storage allocation, paging and file placement on tracks, as well as cutting-stock problems in operations research. Deciding the optimum exactly is NP-hard, so the practical question is how badly simple rules can do. The two simplest on-line rules, First-Fit and Best-Fit, are still the baseline against which every later bin-packing heuristic is measured.

Timeline:

1972. Garey, Graham and Ullman announce that First-Fit uses at most about $1.7$ times the optimal number of bins (Proc. 4th ACM STOC, 1972); Johnson's thesis (MIT, 1973) develops the analysis.
1974. Johnson, Demers, Ullman, Garey and Graham prove $FF(L)\le 1.7L^*+2$ and $BF(L)\le 1.7L^*+2$ for every list, and give lists with $FF(L)=BF(L)>1.7L^*-8$ for every optimum $L^*=k$ , so the asymptotic worst-case ratio of both rules is exactly $\tfrac{17}{10}$ (SIAM J. Comput. 3(4)). This paper is the source of the mission.
1976–2014. The additive constant is lowered: Garey, Graham, Johnson and Yao (1976) show $FF(L)\le\lceil 1.7L^*\rceil$ , and Dósa and Sgall prove the tight bound $FF(L)\le\lfloor 1.7L^*\rfloor$ (STACS 2013) and the same bound for Best-Fit (ICALP 2014).

Setting

A list is a finite sequence $L=(a_1,a_2,\dots,a_n)$ of real numbers in $(0,1]$ ; values may repeat. A bin has capacity $1$ , and its level is the sum of the numbers in it. The optimum $L^*$ is the minimum number of bins into which the elements of $L$ can be placed so that no bin contains numbers whose sum exceeds $1$ .

Both rules place $a_1,\dots,a_n$ in this order into bins $B_1,B_2,\dots$ , each initially at level $0$ , and never move an element once placed.

First-Fit (FF) places $a_i$ into the bin $B_j$ of least index whose level $\beta$ satisfies $\beta\le 1-a_i$ .
Best-Fit (BF) places $a_i$ into a bin whose level $\beta$ satisfies $\beta\le 1-a_i$ and is as large as possible, taking the least index among ties.

$FF(L)$ and $BF(L)$ are the numbers of nonempty bins at the end. The worst-case ratio at optimum $k$ is

R_{FF}(k)=\sup\Bigl\{\frac{FF(L)}{L^*}:L^*=k\Bigr\},\qquad R_{BF}(k)=\sup\Bigl\{\frac{BF(L)}{L^*}:L^*=k\Bigr\}.

The analysis also uses a weighting function $W:[0,1]\to[0,1]$ , piecewise linear with $W(\alpha)=\tfrac65\alpha$ on $[0,\tfrac16]$ , $\tfrac95\alpha-\tfrac1{10}$ on $(\tfrac16,\tfrac13]$ , $\tfrac65\alpha+\tfrac1{10}$ on $(\tfrac13,\tfrac12]$ and $1$ on $(\tfrac12,1]$ , and the coarseness of a bin of a completed packing: the largest $1-\operatorname{level}(B')$ over the bins $B'$ of smaller index, and $0$ for the first bin.

Formalization targets

Goal: the asymptotic ratio (Corollary of Section 2, p. 306)

\lim_{k\to\infty}R_{FF}(k)=1.7\qquad\text{and}\qquad\lim_{k\to\infty}R_{BF}(k)=1.7.

The goal fixes only the asymptotic ratio and leaves the additive constants free, so it is the statement that survives the later improvements of the constants.

Milestones, in the order the proof uses them

Claim 2.2.1 (p. 304): a bin with total size at most $1$ has $\sum_i W(b_i)\le\tfrac{17}{10}$ .
Claim 2.2.2 (p. 305): in an FF or BF packing, every element placed into a bin before the bin was more than half full exceeds the bin's coarseness.
Claim 2.2.3 (p. 305): a bin of coarseness $\alpha<\tfrac12$ whose level exceeds $1-\alpha$ has weight at least $1$ .
Claim 2.2.4 (p. 306): a bin of coarseness $\alpha<\tfrac12$ with weight $1-\beta$ , $\beta>0$ , either holds a single element at most $\tfrac12$ or has level at most $1-\alpha-\tfrac59\beta$ .
Theorem 2.2 (p. 304): $FF(L)\le 1.7L^*+2$ and $BF(L)\le 1.7L^*+2$ for every list.
Theorem 2.1 (p. 301): for every $k\ge1$ there is a list with $L^*=k$ and $FF(L)=BF(L)>1.7L^*-8$ .

A companion item, not a milestone, records the explicit list of Fig. 3 (p. 307) with $L^*=10$ and $FF(L)=BF(L)=17$ .

Significance

The result fixes the worst-case behaviour of the two simplest bin-packing heuristics: neither ever uses more than about $70\%$ more bins than an optimal packing, and both can be forced to. The weighting-function technique introduced for this bound became the standard method for analysing bin-packing heuristics, including First-Fit Decreasing, Harmonic-type algorithms and on-line lower bounds, and the constant $\tfrac{17}{10}$ is the reference point for later on-line algorithms.

The theorem is proved, and its constants have since been sharpened. No machine-checked proof of any of these results is known. This mission produces a Lean model of on-line bin packing (the optimum, the First-Fit and Best-Fit runs with their placement history, and the worst-case ratio) that the other missions of this paper and later bin-packing formalizations can reuse. It also produces formal proofs of the weighting-function bounds, of the $1.7L^*+2$ upper bound and of the lower-bound construction.

Difficulty

The first idea, charging each bin its level, gives only $FF(L)\le 2L^*+1$ : at most one bin is at most half full. The ratio $\tfrac{17}{10}$ comes from bins that are more than half full but far from full, and a bound on the total size of the elements cannot see them. No property of the final packing alone suffices: the bins that are far from full can only be controlled through the order in which the rule opened and filled them, so the argument depends on the dynamics of the run. On the lower-bound side, the natural periodic list (sizes near $\tfrac16,\tfrac13,\tfrac12$ , p. 301) gives only the ratio $\tfrac53$ ; reaching $\tfrac{17}{10}$ needs a list on which both rules waste space in every medium bin, for every $k$ , while $L^*$ is still known exactly.

Formalization scope

A list is L : List ℝ with the hypothesis IsList L (every element in $(0,1]$ ), and every statement assumes it. $L^*$ is optBins L, the least b : ℕ for which some assignment Fin L.length → Fin b has every bin sum at most $1$ . A run is a fold over the list that keeps only the nonempty bins, in index order, each with its contents in placement order. A new bin is opened at the end exactly when no nonempty bin fits, which is the paper's "least $j$ " over infinitely many initially empty bins, since elements are positive. The fit test is the non-strict $\beta+a_i\le1$ , and Best-Fit breaks ties by least index. The placement history (the bin chosen for each element and that bin's level just before) is read off the run on the prefix of the list. Indices are $0$ -based. Coarseness is computed in the completed packing. $W$ is a function ℝ → ℝ and is only ever applied to elements of $(0,1]$ . $R_{FF}(k)$ and $R_{BF}(k)$ are suprema in the extended nonnegative reals $[0,\infty]$ , and the limit is taken there.

A real-valued supremum would be $0$ on an empty or unbounded family, and the limit statement would then say nothing about the algorithms. The extended-real supremum rules this trivialization out. Every claim is stated for the concrete First-Fit run and the concrete Best-Fit run, not for an abstract rule with the properties used in the proof.

Claim 2.2.4 is printed with alternative (i) " $m=1$ and $b_1<\tfrac12$ ", which is false: First-Fit on $(0.6,0.5)$ gives a counterexample. The mission states it with $b_1\le\tfrac12$ , which is what the paper's proof establishes and what the main proof uses. The milestone text keeps the printed version.

The model definitions are reusable for any on-line bin-packing rule, since the run is parameterized by the choice rule. Contributions welcome: proofs of the milestones, general lemmas about the runs (levels stay at most $1$ , at most one bin is at most half full, the history determines the final packing), and the computation of $L^*$ for the explicit lists of Theorem 2.1 and Fig. 3.

Selected references

D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, R. L. Graham, Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms, SIAM Journal on Computing 3(4):299–325, 1974. https://doi.org/10.1137/0203025
M. R. Garey, R. L. Graham, J. D. Ullman, Worst-case analysis of memory allocation algorithms, Proc. 4th ACM STOC, 1972.
D. S. Johnson, Near-Optimal Bin Packing Algorithms, PhD thesis, MIT, 1973.
M. R. Garey, R. L. Graham, D. S. Johnson, A. C. Yao, Resource constrained scheduling as generalized bin packing, J. Combinatorial Theory Ser. A 21, 1976.
G. Dósa, J. Sgall, First Fit bin packing: A tight analysis, STACS 2013, LIPIcs 20:538–549. https://doi.org/10.4230/LIPIcs.STACS.2013.538
G. Dósa, J. Sgall, Optimal analysis of Best Fit bin packing, ICALP 2014, LNCS 8572.

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Graph Theory Linear Optimization Operations Research·Captain: mikedeng1

Finding Minimum-Cost Circulations by Canceling Negative Cycles: Polynomial Termination of Minimum-Mean Cycle CancelingResearch Paper

Motivation

The minimum-cost circulation problem is a central problem of network optimization: transportation, assignment, shortest-path and maximum-flow problems are all special cases, and it is one of the few classes of linear programs with fast combinatorial algorithms. The oldest algorithm for it, the cycle-canceling algorithm of Klein (1967), repeatedly finds a residual cycle of negative cost and pushes as much flow as possible around it. With an arbitrary choice of cycle it can take exponentially many iterations even on integer data, and it need not terminate at all when capacities are irrational.

Goldberg and Tarjan (J. ACM 36(4), 1989) showed that one simple selection rule repairs this: always cancel a residual cycle whose mean cost (cost divided by number of arcs) is as small as possible. The resulting algorithm is strongly polynomial: its number of iterations is bounded by a polynomial in the number of vertices and arcs alone, independent of the magnitudes of capacities and costs. This mission formalizes that bound.

Timeline:

1967, Klein: the cycle-canceling algorithm, without an iteration bound.
1972, Edmonds and Karp: the first polynomial algorithm for minimum-cost flow (capacity scaling), polynomial in the bit length of the capacities.
1985, Tardos: the first strongly polynomial algorithm, introducing the arc-fixing idea that Theorem 3.8 generalizes.
1987–1989, Goldberg and Tarjan: generalized cost scaling and ε-optimality; in this paper, minimum-mean cycle canceling terminates after O(nm² log n) iterations for real costs (Theorem 3.9) and O(nm log(nC)) for integer costs bounded by C (Theorem 3.7).

Setting

A circulation network is a finite directed graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ arcs, which is symmetric ( $(v,w)\in E$ iff $(w,v)\in E$ , so $m$ counts both directions), together with real capacities $u(v,w)$ and real costs $c(v,w)$ , the cost being antisymmetric: $c(v,w)=-c(w,v)$ .

A circulation is a real function $f$ on arcs satisfying $f(v,w)\le u(v,w)$ , $f(v,w)=-f(w,v)$ on every arc, and conservation $\sum_{v:(w,v)\in E} f(v,w)=0$ at every vertex $w$ . Its cost is $\operatorname{cost}(f)=\tfrac12\sum_{(v,w)\in E}c(v,w)f(v,w)$ , and $f$ is minimum-cost (optimal) if no circulation has smaller cost.

The residual capacity of an arc is $u_f(v,w)=u(v,w)-f(v,w)$ ; arcs with $u_f>0$ are residual arcs. A residual cycle is a simple cycle of residual arcs; its capacity is the minimum residual capacity along it, its cost $c(\Gamma)$ is the sum of its arc costs, and its mean cost is $c(\Gamma)/|\Gamma|$ . Canceling a residual cycle raises the flow on each of its arcs by its capacity (and lowers the flow on each reverse arc by the same amount).

The minimum-mean cycle-canceling algorithm starts from any circulation and, while some residual cycle has negative cost, cancels a residual cycle whose mean cost is minimum among all residual cycles. Ties are broken arbitrarily, so the algorithm is a nondeterministic process; a run of length $K$ is any sequence $f_0,\dots,f_K$ of circulations produced by $K$ such iterations.

The analysis uses a price function $p:V\to\mathbb R$ , the reduced cost $c_p(v,w)=c(v,w)+p(v)-p(w)$ , and ε-optimality: for $\varepsilon\ge0$ , $f$ is ε-optimal if some $p$ gives $c_p(v,w)\ge-\varepsilon$ on every residual arc. The quantity $\varepsilon(f)$ is the least such $\varepsilon$ , and an arc is ε-fixed if all ε-optimal circulations carry the same flow on it.

Formalization targets

Goal: Theorem 3.9, with the proof's constant

For every circulation network with $n\ge2$ vertices, $m$ arcs, arbitrary real capacities and arbitrary real antisymmetric costs, every run of the minimum-mean cycle-canceling algorithm has length

K\ \le\ n\,m^2\,\lceil \ln n+1\rceil .

The statement quantifies over all starting circulations, all tie-breaking choices and all real data; it is the paper's $O(nm^2\log n)$ with the constant its proof establishes.

Milestones

In the order the proof uses them: Theorem 2.1 (optimal iff no negative residual cycle), Theorem 3.1 (optimal iff some price function has $c_p\ge0$ on residual arcs), Theorem 3.3 ( $\varepsilon(f)=-\mu(f)$ for nonoptimal $f$ , where $\mu(f)$ is the minimum cycle mean of the residual graph), Lemma 3.5 (a minimum-mean cancellation does not increase $\varepsilon(f)$ ), Lemma 3.6 ( $m$ cancellations shrink $\varepsilon(f)$ by a factor $1-1/n$ ), and Theorem 3.8 (an arc with $|c_p(v,w)|\ge2n\varepsilon$ is ε-fixed).

Significance

Theorem 3.9 shows that a classical, natural algorithm is strongly polynomial: its iteration count depends only on the combinatorial size of the network. Combined with Karp's $O(nm)$ minimum-mean cycle algorithm it yields an $O(n^2m^3\log n)$ strongly polynomial algorithm (Theorem 3.10), and its method, measuring progress by the minimum cycle mean and fixing arcs once $\varepsilon(f)$ is small, underlies the faster cancel-and-tighten algorithm of Section 4 and later strongly polynomial analyses of network-flow and related algorithms.

The theorem has been proved since 1989; this mission's contribution is a machine-checked proof. To the best of the platform's catalogue, no cycle-canceling bound, minimum cycle mean or ε-optimality statement has been formalized. The platform does hold the negative-cycle optimality criterion in a different model (LinearOptimization.network_no_negative_cycle_optimal, Bertsimas–Tsitsiklis Theorem 7.6, with nonnegative flows and supplies) and a flow decomposition theorem (LinearOptimization.network_flow_decomposition); both are related to milestones here but are stated for a different network model.

Difficulty

The obvious potential function, the cost of the circulation, decreases at every iteration but by amounts that depend on the data, so it yields no bound independent of the capacities and costs. The analysis instead has to track $\varepsilon(f)$ , an infimum over price functions, and relate it to the minimum cycle mean of a residual graph that changes after each cancellation, including arcs that appear only because of earlier cancellations. The strongly polynomial part needs a second ingredient: showing that the flow on some arc never changes again, which requires comparing the current circulation with all other ε-optimal circulations of the network, not only those the algorithm visits.

Formalization scope

Vertices form a finite type V; the arc set is E : Finset (V × V); capacities, costs and flows are real functions V → V → ℝ read only on E. $n$ is Fintype.card V and $m$ is E.card, counting $(v,w)$ and $(w,v)$ separately, as in the paper. Cycles are nonempty duplicate-free vertex lists, whose arcs are the cyclically consecutive pairs; one- and two-vertex cycles are allowed and have cost $0$ . Minimum mean is taken over all residual simple cycles of the current circulation. $\varepsilon(f)$ is an infimum (sInf) over a set that is nonempty and bounded below for every circulation; its attainment is to be proved, never assumed.

Explicit constants replacing the paper's $O(\cdot)$ :

Theorem 3.9: the paper prints $O(nm^2\log n)$ ; its proof uses groups of $k=m\,n\lceil\ln n+1\rceil$ iterations, at most $m$ of them, so the goal states $K\le n\,m^2\lceil\ln n+1\rceil$ with the natural logarithm.
The standing assumption $n\ge2$ (p. 874) is kept on the goal; the standing assumption $m\ge n$ is not used by the proof and is omitted.

"Terminates after at most $B$ iterations" means that every run has length at most $B$ . Asserting only that some run is short, or that the process eventually stops, does not formalize the theorem; nor does a step relation that drops negativity, simplicity of the cycle, minimality of the mean over all residual cycles, or the update by exactly the cycle's capacity.

A complete development needs cycle decomposition of the difference of two circulations, LP duality for circulations (Theorem 3.1), and bookkeeping for the residual graph under cancellation. These are reusable for any cycle-canceling or cost-scaling analysis, and contributions of that infrastructure as separate lemmas are welcome. Theorem 3.7 (the integer-cost bound) and Section 4 are outside this mission.

Selected references

A. V. Goldberg, R. E. Tarjan, Finding Minimum-Cost Circulations by Canceling Negative Cycles, J. ACM 36(4):873–886, 1989. https://doi.org/10.1145/76359.76368
M. Klein, A primal method for minimal cost flows with applications to the assignment and transportation problems, Management Science 14(3):205–220, 1967. https://doi.org/10.1287/mnsc.14.3.205
É. Tardos, A strongly polynomial minimum cost circulation algorithm, Combinatorica 5(3):247–255, 1985. https://doi.org/10.1007/BF02579369
A. V. Goldberg, R. E. Tarjan, Finding minimum-cost circulations by successive approximation, Mathematics of Operations Research 15(3):430–466, 1990. https://doi.org/10.1287/moor.15.3.430
R. M. Karp, A characterization of the minimum cycle mean in a digraph, Discrete Mathematics 23(3):309–311, 1978. https://doi.org/10.1016/0012-365X(78)90011-0
J. Edmonds, R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM 19(2):248–264, 1972. https://doi.org/10.1145/321694.321699

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Operations Research Optimization·Captain: mikedeng1

An Optimal On-Line Algorithm for Metrical Task System 1: Every n-State Metrical Task System Has Competitive Ratio 2n - 1Research Paper

Motivation

A system that processes a stream of tasks can often be configured in several ways, and the configuration affects both the cost of the current task and the cost of switching before the next one: paging schemes, replicated files, server placements. When the future is unknown, the natural worst-case yardstick is competitive analysis, introduced by Sleator and Tarjan for list update and paging (Sleator–Tarjan 1985): an on-line strategy is compared with the optimal strategy that knows the whole input in advance.

Borodin, Linial and Saks (J. ACM 1992; conference version STOC 1987) proposed metrical task systems as a single model containing all such problems, and determined the exact deterministic competitive ratio of every such system. Their theorem is the starting point of the on-line-algorithms literature on metrical task systems, the $k$ -server problem (Manasse–McGeoch–Sleator 1990) and their randomized variants.

Timeline. 1985: Sleator and Tarjan introduce competitive analysis for paging and list update. 1987: Borodin, Linial and Saks prove $w(S,d)=2n-1$ for every $n$ -state metrical task system (journal version 1992). 1990: Manasse, McGeoch and Sleator extend the task-system model to restricted task sets and pose the $k$ -server conjecture. The randomized ratio of the uniform task system, bounded in the same paper between $H(n)$ and $2H(n)$ , is the subject of the companion mission.

Setting

A task system $(S,d)$ has a finite set $S$ of $n$ states and a transition-cost matrix $d$ with $d(i,i)=0$ , $d(i,j)>0$ for $i\neq j$ , and the triangle inequality $d(i,j)+d(j,k)\ge d(i,k)$ . It is metrical if also $d(i,j)=d(j,i)$ .

A task $T$ is a vector of nonnegative processing costs $T(s)$ , $s\in S$ . Given a task sequence $\mathbf T=T^1\cdots T^m$ and an initial state $s_0$ , a schedule is a map $\sigma:\{0,\dots,m\}\to S$ with $\sigma(0)=s_0$ ; task $T^i$ is processed in state $\sigma(i)$ , and the cost is

c(\mathbf T;\sigma)=\sum_{i=1}^m d(\sigma(i-1),\sigma(i))+\sum_{i=1}^m T^i(\sigma(i)).

The off-line optimum $c_0(\mathbf T)$ is the minimum over all schedules. An on-line algorithm $A$ chooses $\sigma(i)$ knowing only $s_0$ and $T^1,\dots,T^i$ ; its cost is $c_A(\mathbf T)$ . For $w>0$ , $A$ is $w$ -competitive if there is a constant $K_w$ with $c_A(\mathbf T)\le w\,c_0(\mathbf T)+K_w$ for every finite task sequence. The competitive ratio of $A$ is $w(A)=\inf\{w: A\text{ is }w\text{-competitive}\}$ , and the competitive ratio of the task system is $w(S,d)=\inf_A w(A)$ .

For the upper bound the paper also uses continuous-time schedules, in which task $T^i$ occupies the interval $[i,i+1)$ and the scheduler may change state at any real time, paying $\int_i^{i+1}T^i(\sigma(t))\,dt$ for processing. For a general (possibly asymmetric) matrix $d$ , the cycle offset ratio $\psi(d)$ is the maximum over closed walks $s_0,\dots,s_k=s_0$ of $\sum_i d(s_{i-1},s_i)\big/\sum_i d(s_i,s_{i-1})$ ; it equals $1$ when $d$ is symmetric.

Formalization targets

Goal: Theorem 1.1

For every metrical task system $(S,d)$ with $n$ states,

w(S,d)=2n-1 .

The value depends on $n$ only, not on the distances.

Milestones

Lemma 2.1. If $c_0(T^1\cdots T^m)\to\infty$ along an infinite task sequence $\mathbf T$ , then $w(A)\ge w_{\mathbf T}(A)=\limsup_m c_A/c_0$ .
Theorem 2.2. Against the cruel taskmaster $M(\varepsilon)$ , which charges $\varepsilon$ in the state the algorithm currently occupies,

w_{\mathbf T(\varepsilon)}(A)\ge\frac{2n-1}{1+\varepsilon/\min_{i\neq j}d(i,j)} .

Lemma 3.1. Every on-line continuous-time algorithm is matched, on every task sequence, by an on-line discrete-time algorithm.
Lemmas 6.3, 6.4, 6.2. Properties of the functions $f_k$ that drive the algorithm $A^*_d$ : $f_k(s)-f_k(s')\le d(s',s)$ ; the identity $2\sum_{s\ne s_k}f_k(s)+f_k(s_k)=C_{k-1}+\sum_{i\le k}d(s_i,s_{i-1})$ ; and $f_k\le h_k$ , the off-line cost at the $k$ -th transition time.
Theorem 6.1 (= Theorem 1.2). For every task system, symmetric or not, $A^*_d$ has competitive ratio at most $(2n-1)\psi(d)$ .

Significance

The theorem settles the deterministic competitive ratio of the whole class of metrical task systems: the lower bound says that no deterministic on-line strategy can beat $2n-1$ on any metric, and the upper bound supplies one algorithm that achieves it on every metric. For asymmetric costs the same algorithm gives $(2n-1)\psi(d)$ . The $2n-1$ lower bound is also the benchmark against which restricted models, such as paging and the $k$ -server problem, measure their improvements, and the randomized question it leaves open drove much of the later work on metrical task systems.

The result was proved in 1987 and is standard; to the best of our knowledge no machine-checked proof exists. A formal development would provide a reusable model of deterministic on-line algorithms and competitiveness (on-line maps from task prefixes, additive competitiveness, infima over algorithms), an adversary construction by mutual recursion with an arbitrary algorithm, and an exact treatment of continuous-time schedules with piecewise-constant task costs. These pieces are reusable for other competitive-analysis results.

Difficulty

The lower bound is not a single bad input: the adversary is built from the algorithm it plays against, so the hard task sequence exists only as a recursion interleaved with the algorithm's choices, and the bound must hold for every deterministic on-line map, including ones that behave erratically. Obtaining the exact constant $2n-1$ , rather than some $\Omega(n)$ bound, requires a sharp estimate of the off-line cost of that sequence.

The upper bound needs an algorithm defined in continuous time, whose transition times are determined by accumulated processing costs; the budgets can be zero, so transitions can be instantaneous, and a formal cost must remain well defined before one knows that only finitely many transitions occur. Relating the off-line cost function at those times to the recursively defined $f_k$ (Lemma 6.2) requires reasoning about all continuous-time off-line schedules. Finally, the goal combines both directions through infima over all on-line algorithms, and the discretization of Lemma 3.1 must be composed with the continuous-time algorithm.

Formalization scope

States form a finite type S (Fintype, DecidableEq, Nonempty); the goal is stated for all $n\ge1$ , where $n=1$ gives $w(S,d)=1$ . Task costs are finite nonnegative reals; the paper also allows $+\infty$ entries, which are excluded (this affects neither bound). A task sequence is T : Fin m → S → ℝ, with T i the paper's $T^{i+1}$ , and a schedule is σ : Fin (m+1) → S. An on-line algorithm is a map sending $(s_0,[T^1,\dots,T^i])$ to $\sigma(i)$ , so on-line behaviour is built into the type. Competitiveness is written additively, $c_A\le w\,c_0+K$ , with $K$ independent of the task sequence and of $s_0$ .

The competitive ratio competitiveRatio d is the real infimum of the set of all $w$ for which some on-line algorithm is $w$ -competitive. It is not defined as an infimum of per-algorithm real infima: a non-competitive algorithm has $W_A=\emptyset$ , whose real infimum is $0$ , and that would drag $w(S,d)$ to $0$ for every system. Since the goal's value $2n-1$ is at least $1$ while the empty set's real infimum is $0$ , the goal cannot hold vacuously.

Continuous-time algorithms are given as lists of $(\text{state},\text{length})$ pieces per unit interval; processing integrals are exact finite sums. The algorithm $A^*_d$ minimizes over states different from the current one, as its proof requires (the printed rule ranges over all states, and would stall); ties are left arbitrary. Its budgets may be $0$ , its entry times are Option ℝ, and its cost is a sum in $[0,\infty]$ , so that Theorem 6.1 itself asserts that only finitely many transitions occur. The ratio $\psi(d)$ excludes closed walks that never move, and Theorems 2.2 and 6.1 require $n\ge2$ , where $\min_{i\ne j}d(i,j)$ and $\psi(d)$ are defined. Lemma 3.1 is stated comparing $A$ with $A'$ (the printed statement says "as well as $A$ ").

A complete development needs: the discrete model and off-line optimum (finite minimum over schedules), limsup arguments in EReal, continuous-time schedules with piecewise-constant costs, and the recursion defining $A^*_d$ . Proofs of any milestone, including the purely combinatorial Lemmas 6.3 and 6.4, are welcome, as is a formal composition of Lemma 3.1 with Theorem 6.1.

Selected references

A. Borodin, N. Linial, M. E. Saks, An optimal on-line algorithm for metrical task system, Journal of the ACM 39(4):745–763, 1992. https://doi.org/10.1145/146585.146588
D. D. Sleator, R. E. Tarjan, Amortized efficiency of list update and paging rules, Communications of the ACM 28(2):202–208, 1985. https://doi.org/10.1145/2786.2793
M. S. Manasse, L. A. McGeoch, D. D. Sleator, Competitive algorithms for server problems, Journal of Algorithms 11(2):208–230, 1990. https://doi.org/10.1016/0196-6774(90)90003-W

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Analysis Machine Learning Probability·Captain: MiltMont

Flow Matching Theorem 1: Marginal Continuity EquationResearch Paper

From conditional motion to a marginal probability path

Flow matching models a changing probability distribution using a time-dependent velocity field. A conditional model specifies a density and a velocity separately for each conditioning point. The mathematical question is whether those conditional descriptions determine a velocity for the mixture distribution. This mission concerns the continuity-equation formulation of Theorem 1 of Lipman, Chen, Ben-Hamu, Nickel, and Le, Flow Matching for Generative Modeling (ICLR 2023). The source is arXiv:2210.02747v2, Section 3.1 and Appendix A.

Densities, velocities, and probability flux

Fix a natural number $d$ and let $E=\mathbb R^d$ . The conditioning distribution $Q$ is a Borel probability measure on $E$ . At time $t$ , position $x$ , and conditioning point $z$ , write $\rho(t,x,z)$ for the conditional density and $v(t,x,z)\in E$ for the conditional velocity. The variable $x$ is integrated against Lebesgue measure; $z$ is integrated against $Q$ . These roles remain distinct even though both variables take values in the same space.

The conditional flux is $F(t,x,z)=\rho(t,x,z)v(t,x,z)$ . The marginal density, marginal flux, and marginal velocity are defined by

p(t,x)=\int_E\rho(t,x,z)\,dQ(z),\qquad J(t,x)=\int_E F(t,x,z)\,dQ(z),\qquad u(t,x)=p(t,x)^{-1}J(t,x).

These definitions express equations (6) and (8) using a probability measure rather than a data-density function. This representation also allows discrete conditioning distributions. Every conditional density is strictly positive and normalized on $0\leq t\leq1$ , jointly measurable in $(x,z)$ , and integrable in $z$ at each fixed $(t,x)$ .

The divergence of a differentiable vector field is the sum of the diagonal entries of its derivative. A density and velocity satisfy the classical continuity equation when their flux is spatially differentiable and the density has time derivative equal to minus that divergence.

Formalization targets

The goal asserts that $p(t,\cdot)$ is a positive probability density for every $t\in[0,1]$ and that

\partial_t p(t,x)+\operatorname{div}_x\bigl(p(t,x)u(t,x)\bigr)=0\qquad(0<t<1,\ x\in E).

The hypotheses require the conditional continuity equation for $Q$ -almost every conditioning point, at each interior time and spatial point. They also specify a sufficient local domination package for differentiation under the integral. This is an explicit classical interpretation of the regularity qualification in the proof of Theorem 1.

Four supporting targets isolate the mathematical assertions used by this formulation: the probability-density property of equation (6); time differentiation under the conditioning integral; spatial divergence under the conditioning integral; and the velocity/flux identity corresponding to equation (8). The source contains these equations and operations rather than separately numbered supporting lemmas, so the milestone titles identify the relevant equation or proof passage.

What completing the formalization provides

The deliverable is a checked interface for passing from a measurable family of conditional continuity equations to the continuity equation of its mixture. It records which variables are differentiated, which measure is used for averaging, where positivity is needed, and which assumptions justify each analytic operation. The time and spatial differentiation lemmas are stated for general measures and integrands, making them reusable outside this particular probability model.

The mathematical result is already proved in the cited paper. The uploaded theorem items are open formalization targets, with explicit proof placeholders. Successful local compilation checks their types and imports; it does not establish their conclusions. The definition module contains no proof placeholders.

Analytic obligations

Pointwise differentiability of every conditional function does not by itself justify differentiating an integral over the conditioning variable. The regularity predicates therefore require a neighborhood independent of that variable, an integrable bound for the derivative norm throughout that neighborhood, and almost-everywhere measurability of the integrand and derivative. Time and space receive separate predicates because their derivatives take values in different spaces.

There is also a distinction between density normalization in $x$ and integrability in $z$ at a fixed position. The formal assumptions record both. A probability measure on the conditioning space does not make every measurable function integrable. These conditions prevent the totalized Bochner integral from silently supplying a default value where an intended integral fails to exist.

Formalization scope

Space is represented by Fin d → ℝ, with its standard finite-product Borel structure and Lebesgue measure. Its norm is the standard product norm used by mathlib. All finite dimensions, including dimension zero, are included. Time-dependent functions are defined on all real times, while density assumptions apply on the closed unit interval and derivative conclusions apply on its interior. No endpoint time derivative is asserted.

The regularity package is one sufficient realization of the source's Leibniz-rule assumption, not a claim to the weakest possible hypotheses. Conditional continuity equations may hold almost everywhere in the conditioning variable; their exceptional sets may depend on the fixed time and position. Spatial differentiability of the marginal flux is part of the conclusion, so the equation cannot be satisfied merely through the default value of an undefined derivative.

The target is the PDE formulation. It does not assert existence of a global ODE flow, uniqueness of transported measures, or equality with a flow pushforward. Those require a separate transport development. It also asserts no endpoint approximation to a data distribution and no theorem about optimization, neural networks, or Gaussian paths. No marginal continuity equation or differentiation–integration interchange is assumed as an input.

Required infrastructure consists of Bochner integration, finite-dimensional differentiation, finite sums of derivative coordinates, and product-measure integration. Contributions may prove the supporting targets or the goal directly while preserving their statements and the distinction between classical PDE and flow-transport claims.

Selected references

Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow Matching for Generative Modeling. ICLR 2023. arXiv:2210.02747v2, Section 2, Section 3.1, Theorem 1, equations (6), (8), and (26), and Appendix A's proof of Theorem 1.
mathlib contributors. ParametricIntegral.lean, revision 0df444a360eaa60ab8c11dca51a86af692955474. Differentiation under the integral.

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Information Theory·Captain: Lucas

Shannon 1949: Perfect SecrecyResearch Paper

Motivation

Cryptography before 1949 was a catalogue of ciphers and of the tricks that broke them. C. E. Shannon's Communication Theory of Secrecy Systems (Bell System Technical Journal 28(4):656–715, 1949) replaced the catalogue with a probabilistic model: a cipher is a family of invertible maps from messages to cryptograms, indexed by a key drawn from a known distribution, and the cryptanalyst's state of knowledge after an interception is the a posteriori distribution over messages. Part II of that paper asks when interception conveys nothing at all. The answer — perfect secrecy — is the origin of the one-time pad's security proof and of the modern habit of defining security as the indistinguishability of a posteriori from a priori beliefs.

This mission formalizes §10, Perfect Secrecy: the definition, the necessary and sufficient condition (Shannon's Theorem 6), the counting bound that the key set be at least as large as the message set, the cyclic system that attains the bound, the Latin-square description of the systems that attain it, and the entropy form of the constraint.

Setting

A finite secrecy system consists of three finite sets — the messages $M$ , the keys $K$ , the cryptograms $E$ — together with an enciphering map $T_k : M \to E$ for each key $k$ . Each $T_k$ is non-singular, i.e. injective, so a receiver who knows the key deciphers unambiguously. The key is drawn from an a priori distribution $P(k) \ge 0$ , $\sum_k P(k) = 1$ , and the message from an a priori distribution $P(M) \ge 0$ , $\sum_M P(M) = 1$ , independently of the key.

Three derived quantities carry the theory. The key weight

P_M(E) \;=\; \sum_{k \,:\, T_k M = E} P(k)

is the probability that the cryptogram is $E$ given that the message is $M$ . The cryptogram probability is $P(E) = \sum_M P(M) P_M(E)$ , the probability of obtaining $E$ from any cause. The a posteriori probability of the message after interception is $P_E(M) = P(M) P_M(E) / P(E)$ , defined for cryptograms with $P(E) \neq 0$ .

A system has perfect secrecy when, for every a priori message distribution and every cryptogram that can occur, $P_E(M) = P(M)$ for all $M$ : interception leaves the cryptanalyst's probabilities unchanged. Quantifying over all a priori message distributions is Shannon's requirement that the equality hold "independently of the values of $P(M)$ ".

Finally, $H(M) = -\sum_M P(M)\log P(M)$ and $H(K) = -\sum_K P(K)\log P(K)$ are the entropies of the message and key choices.

Formalization targets

Goal — Theorem 6

\text{perfect secrecy} \iff P_M(E) = P(E) \quad \text{for all } M, E .

Equivalently, $P_M(E)$ does not depend on $M$ : the total probability of the keys carrying $M_i$ to a given $E$ is the same as that of the keys carrying $M_j$ to the same $E$ . The goal fixes nothing about the sizes of the three sets, and no structure on the key distribution beyond normalization.

Milestone — the Bayes relation (p. 680)

P(E) = \sum_M P(M)P_M(E), \qquad P_E(M) = \frac{P(M)P_M(E)}{P(E)} .

Milestone — the key-counting bound (p. 681)

\text{perfect secrecy} \implies \#M \le \#K .

Milestone — attainability, the cyclic system of Fig. 5 (p. 681)

T_i M_j = E_s, \quad s = i + j \bmod n, \quad P(k) = \tfrac1n \;\Longrightarrow\; \text{perfect secrecy}.

Milestone — Latin-square characterization (p. 681)

\text{perfect secrecy} \ \wedge\ \#M = \#K = \#E \implies \big(\forall M, E\ \exists! k,\ T_k M = E\big) \ \wedge\ P \text{ is uniform}.

Milestone — the entropy constraint (p. 682)

\text{perfect secrecy} \implies H(M) \le H(K) .

Significance

Theorem 6 is the structural fact behind every later statement in the section: the counting bound, the Latin-square description, and the entropy constraint are all read off from the independence of $P_M(E)$ from $M$ . The counting bound and the entropy constraint are the precise sense in which unconditional secrecy is expensive — the key must be at least as uncertain as the message — and they are the reason practical cryptography moved to computational assumptions. The cyclic system supplies the matching construction, so the bound is sharp.

These results are classical and have textbook proofs; what this mission produces is a machine-checked development of them from a single explicit model of a finite secrecy system, including the model itself. Mathlib has extensive measure-theoretic probability and some information theory, but no formalization of Shannon's secrecy systems, of perfect secrecy, or of the results of §10. The shared definition layer — ciphers with injective enciphering maps, key weights, a posteriori probabilities, the perfect-secrecy predicate, finite entropy — is reusable for the rest of the paper: the equivocation results of §11–§13, unicity distance, and the ideal-system theorems of §16 all rest on the same objects.

Difficulty

The two directions of Theorem 6 are not symmetric. Sufficiency is a computation. Necessity has to extract, from a statement about a posteriori probabilities that only constrains cryptograms of positive probability and only for one distribution at a time, a statement about key weights alone; the useful instantiation is a distribution giving every message positive mass, and the cryptograms of probability zero must be handled separately rather than ignored.

The counting and Latin-square milestones are finite combinatorics, but the obvious argument — "for each message and cryptogram pick the key joining them" — needs the keys picked to have positive probability, and the model permits keys of probability zero; this is exactly where an argument that looks complete on paper leaves a gap in Lean. The entropy milestone is the hardest of the five: $H(M) \le H(K)$ passes through the conditional entropies $H(M \mid E) \le H(K \mid E)$ , and Mathlib's information-theoretic API is stated for measures rather than for finite weighted sums, so either a bridge or a self-contained development of finite conditional entropy is required.

Formalization scope

Messages, keys and cryptograms are arbitrary finite types with decidable equality on cryptograms; probabilities are real-valued functions with explicit non-negativity and normalization hypotheses rather than PMF, so that all sums are finite sums. A cipher bundles the enciphering maps, their injectivity, and the key distribution. The a posteriori probability is a real quotient, so it evaluates to zero when the cryptogram has probability zero; every statement that mentions it carries the hypothesis that the cryptogram has positive probability. Entropy uses the natural logarithm, with $x \mapsto -x\ln x$ vanishing at $0$ ; all inequalities are therefore in nats and are unaffected by the choice of base.

One trivializing formalization is ruled out explicitly: perfect secrecy is quantified over all a priori message distributions, not over one fixed distribution, and it is not vacuous — the cyclic milestone exhibits a family of ciphers satisfying it for every $n \ge 1$ .

Contributions welcome beyond the milestones: finite conditional entropy and its chain rule as reusable lemmas, the equivalence of the present perfect-secrecy predicate with the statistical-independence formulation of message and cryptogram, and the extension of the model towards §11's equivocation $H_E(M)$ , which is the next result of the paper to formalize.

Selected references

C. E. Shannon, Communication Theory of Secrecy Systems, Bell System Technical Journal 28(4):656–715, 1949. https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal 27(3):379–423, 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

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Optimization·Captain: mikedeng1

The Design of Approximation Algorithms 11: Planar weighted independent-set PTASTextbook

Motivation

A graph records pairs of objects that cannot be chosen together. An independent set is a choice with no conflicting pair. When each vertex has a nonnegative weight, maximum weighted independent set asks for a conflict-free set with the greatest total weight. This model occurs when choices have values and pairwise incompatibilities. On general graphs the optimization problem is difficult; the planar restriction gives a concrete geometric promise under which approximation is possible. Williamson and Shmoys state a polynomial-time approximation scheme for planar maximum independent set as Theorem 10.11 of The Design of Approximation Algorithms, in the author electronic manuscript at PDF/manuscript page 271.

Setting

The instance has vertices labeled by Fin n, an adjacency table edge : Fin n → Fin n → Bool, and a real weight weight i for each vertex. SimpleGraph.fromRel turns the table into an undirected simple graph G. The theorem requires every weight to be nonnegative. A finite set S is feasible when G.IsIndepSet (S : Set (Fin n)) holds. Its value is ∑ i ∈ S, weight i.

Planarity is expressed by HasPlanarDrawing G: vertices are placed at distinct points of the real plane, and every edge is represented by a continuous simple arc. An arc has no vertex in its interior, and distinct edges can meet only at endpoints. This is a concrete local drawing convention for the planar graph promise. The drawing witnesses the hypothesis; it is not passed as part of the algorithm's input.

The computational model is a fixed finite unit-cost RAM program. Its input includes the vertex count, an integer accuracy parameter, a complete adjacency table, and the real weights in named memory cells. Its instructions include natural-number and real addition and subtraction, exact comparisons, random-access loads and stores, branches, jumps, and halt. The machine starts with the instance preloaded and produces a bitmap after the adjacency table. This precise instruction set is a formalization convention: the book states an arithmetic-operation running time but does not specify a machine language.

Formalization targets

For every real $\varepsilon>0$ , let $k=\max(1,\lceil1/\varepsilon\rceil)$ . The goal asserts that a single finite program and fixed positive natural constants $C,d$ work for every planar instance and every vector of nonnegative real weights. The program must halt after a number $t$ of operations satisfying

t+1\le C\,2^{dk}(n+1)^2.

Its bitmap must represent an independent set $A$ , and for every independent set $S$ ,

(1-\varepsilon)\sum_{i\in S}w_i\le\sum_{i\in A}w_i.

Comparison with every feasible $S$ includes an optimal solution. The same program and constants precede all accuracy and instance quantifiers. The $(n+1)^2$ form includes the empty graph without imposing zero running time. The result corresponds to the $O(2^{O(1/\varepsilon)}n^2)$ bound in Theorem 10.11; the formal statement makes the constants and accuracy parameter explicit.

Significance

The result supplies an accuracy-time tradeoff for planar weighted independent set: any fixed positive accuracy has a quadratic dependence on graph size in this operation model, while the accuracy dependence is exponential. It separates the planar setting from the general graph problem. The finite instruction set and input layout make the computational claim inspectable, including what information the program receives and what counts as an operation.

The cited result is known in the textbook. This mission asks for a Lean proof of the packaged statement; the theorem currently has sorry, and no machine construction or correctness proof is claimed. The original local statement was compiled and source reviewed before packaging. That prior result establishes statement validity in its source workspace, while the upload payload is checked separately.

Difficulty

A high-quality independent set in each planar region does not immediately give a high-quality global set: edges across regions can create conflicts, and discarding boundary vertices can lose weight. A proof must coordinate the approximation inequality with a uniform operation bound for one program across all graph sizes, accuracies, and real weight vectors. It must also show the computed bitmap is always independent and that execution reaches an explicit halt instruction within the bound. These requirements rule out treating an optimizer, an embedding, or a best solution as preloaded input.

Formalization scope

The goal uses finite labeled graphs, nonnegative arbitrary real weights, exact real arithmetic, and the concrete drawing predicate above. It does not claim a bit-complexity bound for encoded reals. The RAM starts from a complete adjacency table and weight vector, with no supplied drawing or optimum. Its step function is deterministic; an invalid program counter is stuck, so the theorem explicitly requires a halt instruction. Outputs are bits in designated natural memory cells and are checked for validity before their weighted value is compared.

The packaged definitions are the drawing predicate, instruction type, state, transition, iteration, input layout, and output decoder. They contain no proof axioms or optimization oracle. The rational-input polynomial bit-time fragment in the source workspace is a separate statement and is outside this goal. A complete contribution would construct the finite program and prove its running time and approximation guarantee in the specified model.

Selected references

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Theorem 10.11, PDF/manuscript p. 271; weighted independent-set setting p. 269 and context pp. 270–272. DOI.

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Linear Optimization Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach IX: General Packing-Covering ConstraintsTextbook

Motivation

Chapter 4's framework (formalized in this series' 04-framework mission) solves the online covering-packing pair only in the restricted setting a(i,j) ∈ {0,1}, b(j) = 1 — every constraint is an unweighted "cover me with at least one of these" condition. Chapter 14 delivers the promise made at the very start of the survey (p. 115: "we show how to extend the ideas we present here to handle general (non-negative) values of a(i,j) and b(j)"): fully general non-negative coefficients, normalized so every constraint reads ∑_i a(i,j)x(i) ≥ 1. This mission formalizes both halves of that generalization — the packing scheme (Theorem 14.1, with a matching lower bound, Lemma 14.2, showing an extra additive term is unavoidable) and the covering scheme (Theorem 14.3, the goal).

Setting

Fix a finite set I of primal (covering) variables with positive costs c(i), and a finite set J of dual (packing) variables/covering constraints, with a(i,j) ≥ 0 for every pair (Fig. 14.1). The packing scheme (Section 14.1) is parameterized by a target competitive ratio B > 0: on each new dual variable y(j) and its coefficients a(i,j), the algorithm increases y(j) continuously and each x(i) by an explicit exponential increment function until the new primal constraint is satisfied, achieving B-competitiveness for the packing objective at the cost of an additive O(log(a_i(max)/a_i(min))) term (beyond the multiplicative O(log n)) in how much each dual constraint can be violated — qualitatively different from Chapter 4's purely multiplicative O(log d) bound, and Lemma 14.2 proves this additive term cannot be removed. The covering scheme (Section 14.2) instead works in phases: each phase assumes a doubling lower bound α(r) on OPT and "forgets" its primal/dual variables once the primal cost exceeds α(r), restarting with α(r+1) = 2α(r) — a structurally different mechanism from Chapter 4's direct algorithms, needed because with general coefficients a single monotone run can no longer be analyzed via one potential function alone.

Formalization targets

Theorem 14.3 (the goal, p. 253): for any B > 0, the phase-based covering scheme (each constraint normalized to ∑_i a(i,j)x(i) ≥ 1/B) is competitive with an explicit ratio 8 log(2n)/B, taken directly from the proof's own final displayed chain, 2α(r) ≤ 4α(r-1) ≤ (8 log(2n)/B) Y(r-1) ≤ (8 log(2n)/B) OPT (p. 253-254) — the theorem's own statement only gives O(log n/B), so this explicit constant is this mission's own instantiation from the proof, not an independent derivation and not a transcription of a displayed theorem-level formula (flagged, per this series' explicit-constants rule).

Two milestones, in attack order:

Theorem 14.1 (p. 249): the packing scheme is B-competitive, and violates each dual constraint by at most the book's own exact displayed bound c(i)·2log(1 + n·a_i(max)/a_i(min))/B (Claim (3) — the exact constant the proof establishes, not the theorem headline's O(·) simplification).
Lemma 14.2 (p. 251): a matching lower bound, on the book's own explicit single-constraint instance, showing the additive log(a(max)/a(min)) term of Theorem 14.1 is necessary.

Significance

This chapter is the survey's demonstration that the primal-dual framework's core technique survives its most natural generalization, at the price of an explicit extra term the chapter also proves is unavoidable — a tight characterization, not merely an upper bound. Every other online covering/packing chapter in this survey (set cover, routing, ad-auctions, bounded allocation) is technically a special case of this chapter's general model; Chapter 4's restricted framework is the pedagogical entry point, and this chapter is where the general theory actually lives. No formal development of the general packing-covering problem was found on the platform as of 2026-09-20; this mission is the first.

Difficulty

Two distinct obstacles, mirroring this chapter's own two schemes. First, Theorem 14.1's proof (p. 249-251) establishes its per-round primal/dual derivative inequality via a direct calculus argument (differentiating the explicit increment function) — formalized here as a hypothesis (hX_le_BY) standing for that calculation, not reproduced, since the goal is a faithful statement of the resulting competitive ratio, and the increment function's own exponential form is transcribed in the theorem's docstring but the differentiation itself is out of scope. Second, Theorem 14.3's phase-based mechanism is genuinely stateful across an unbounded number of phases (each phase resets its own primal/dual variables while the LP's actual variables retain the running maximum) — modeling this process explicitly is comparable in complexity to Chapter 13's level-based algorithm, and this mission makes the same scope choice: the mechanism's output (the resulting cost/profit relationship, hX_le_ratio) is taken as a hypothesis standing for the book's own Claims (1) and (3) combined, rather than constructed phase-by-phase.

Formalization scope

GeneralInstance I J bundles Fig. 14.1's fully general LP data (a(i,j) ≥ 0, c(i) > 0) — restated locally (not importing 04-framework's CoveringInstance) per this series' rule against cross-draft imports, even though this chapter is the direct generalization of that one. aMax/aMin are the per-variable (not per-instance) maximum and minimum-non-zero coefficients Theorem 14.1 needs. harmonicNum is restated locally (duplicated from 13-bounded-allocation's own definition, for the same no-cross-draft-import reason). Both goal-adjacent theorems use this series' weak-duality "competitive against any feasible comparison solution" pattern (04-framework, reused as a convention, not re-derived): Theorem 14.1 against any feasible packing comparison (matching that it concerns the packing side), Theorem 14.3 against any feasible covering comparison (matching the covering side). Welcome contributions: completing the three sorrys (Theorem 14.1's calculus argument, Theorem 14.3's phase-based mechanism constructed explicitly, and Lemma 14.2's direct summation argument, which is the most tractable of the three to actually prove), and formalizing the sanity check that both schemes reduce to Chapter 4's Algorithm 1/2/3 when a(i,j) ∈ {0,1}, b(j) = 1 (checked by hand in SELF_REVIEW.md, not as a Lean lemma).

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024

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The Design of Competitive Online Algorithms via a Primal-Dual Approach VIII: The Bounded Allocation ProblemTextbook

Motivation

The classical online allocation (AdWords) problem has a tight 1 - 1/e competitive ratio in general, achieved by the water-level algorithm and matched by a lower bound in which the number of buyers interested in each item can be as large as the total number of buyers. Buchbinder and Naor's Chapter 13 observes that in many realistic settings each item's interested-buyer set is much smaller than the total buyer population, and shows this structural fact — an explicit bound d on interested buyers per item — provably beats 1 - 1/e for every finite d, via a deliberately non-water-level algorithm. This mission formalizes that algorithm's competitive ratio and its matching lower bound.

Setting

A seller offers items to n buyers one at a time; buyer i has budget B(i) > 0. Each item j has a fixed price b(j) > 0 and a set S(j) of interested buyers with |S(j)| ≤ d. The fractional LP relaxation (Fig. 13.1) allocates y(i,j) ∈ [0,1] of item j to buyer i, subject to each item being sold at most once in total and each buyer's spending never exceeding budget; the seller's objective is to maximize total revenue ∑_j ∑_{i∈S(j)} b(j)y(i,j). Buyers are partitioned into d+1 levels by the fraction of budget spent so far (level k = spent between k/d and (k+1)/d); on each new item, the allocation algorithm splits it equally among the interested buyers in the lowest non-empty level, moving to the next level once that level's buyers are exhausted or saturated — deliberately not the naive "water-level" rule of splitting among the least-spent buyers, which the book shows cannot beat 1-1/e even for small d. The analysis tracks a piecewise-linear trade-off potential function f_d, built from a geometric sequence, that relates each buyer's level to their contribution to a feasible primal (covering) solution.

Formalization targets

Theorem 13.1 (the goal, p. 240): the allocation algorithm is C(d)-competitive, with the book's own explicit closed form C(d) = 1 - (d-1)/(d(1+1/(d-1))^{d-1}) — strictly better than 1 - 1/e for every finite d, approaching it as d → ∞ (Table 13.1). Formalized via the survey's standard weak-duality pattern (as in 04-framework's Theorem 4.3): given the algorithm's per-item primal/dual cost changes satisfying the book's core inequality ΔX(j) ≤ (1/C(d))ΔY(j) (established there by a potential-function case analysis, not reproduced here), the algorithm's realized profit is C(d)-competitive against any feasible comparison allocation.

Lemma 13.2 (milestone, p. 244): a matching lower bound, C(d) ≤ 1 - (k - kH(d) + ∑_{i=1}^k H(d-i))/d, where H is the harmonic number and k is the largest value with H(d) - H(d-k) ≤ 1.

Significance

This chapter is the survey's demonstration that a structural restriction invisible to the classical 1-1/e lower bound — a bound on demand concentration, not on budgets or prices — can be exploited algorithmically, and the exploiting algorithm is not the naive generalization of the water-level rule but a genuinely different level-based, "who's-behind" allocation rule. No formal development of the bounded allocation problem was found on the platform as of 2026-09-20; this mission is the first.

Difficulty

The chapter's own proof of Theorem 13.1 (p. 241-244) is a page-and-a-half case analysis on how an item's fractional allocation crosses level boundaries, bookkeeping the change in both the primal potential-function value and the dual profit through several sub-cases (an item fully absorbed by one level; an item that empties a level and continues into the next; a level exhausting every interested buyer's budget). This mission formalizes the resulting per-item inequality ΔX(j) ≤ (1/C(d))ΔY(j) as a hypothesis (the theorem's own headline claim, not a case-by-case re-derivation) rather than modeling the stateful, order-dependent level-allocation process itself — the same scope choice this series makes for Chapter 11's randomized rounding process, where the book similarly omits (there, entirely; here, gives but does not ask this mission to reproduce) the underlying case analysis. The potential function f_d and its connection to the algorithm's primal variable (allocX) are formalized precisely, since they are what the goal's proof and Lemma 13.2 both depend on structurally, even though the case analysis linking them to ΔX/ΔY is left as the theorem's sorry.

Formalization scope

AllocationInstance I J bundles the LP data of Fig. 13.1 (S, B, b, d ≥ 2, ∀j, |S(j)|≤d) — restated locally per this series' rule that concurrent drafts cannot import each other, even though the problem is a special case of Chapter 10's ad-auctions model (the two chapters' algorithms differ: Chapter 10's is proportional-to-remaining-budget, this chapter's is level-based). geomSeq/potential transcribe the geometric sequence a_t and the potential function f_d at its level grid points exactly (not extended to non-grid-point reals, since Theorem 13.1's and Lemma 13.2's own statements only need the grid values). allocX connects the potential function to the algorithm's primal variable via each buyer's final level t(i). packingFeasible/packingValue transcribe Fig. 13.1's dual/packing LP with a genuine two-index allocation y : I → J → ℝ (not collapsed to a single per-item variable, unlike Chapter 4's simpler 0/1-coefficient framework). harmonicNum is the ordinary harmonic number. The level-based algorithm's literal stateful per-item update rule (which buyers move between which levels, in what order, within a single item's allocation) is not modeled directly — a documented scope reduction (STATUS.md), not a substitution of the "water-level" algorithm the book explicitly warns against (this mission's theorem13_1 commits to neither algorithm's literal rule, only to the resulting invariant the book's own proof establishes for the level-based one). Welcome contributions: completing the two sorrys, and modeling the level-allocation process explicitly enough to derive hinvariant from first principles.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024
B. Kalyanasundaram, K. Pruhs. An optimal deterministic algorithm for online b-matching. Theoretical Computer Science, 233(1-2):319-325, 2000 (cited as [73], the 1-1/e lower bound).

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Graph Theory Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach VII: Online Group Steiner TreesTextbook

Motivation

The group Steiner tree problem generalizes the ordinary Steiner tree problem: given a rooted tree and several groups of vertices, find a minimum-cost subtree that connects at least one vertex of each group to the root. It is a canonical instance of the generalized-connectivity family this survey studies in Chapter 11 — a family that also contains the online set-cover problem (Chapter 5 of this series) as a special case. Buchbinder and Naor's chapter shows how to convert the celebrated offline randomized-rounding algorithm of Garg, Konjevod and Ravi [56] into an online one, by imitating its per-edge coupling structure one iteration at a time as the online fractional solution (obtained from this survey's own Chapter 4 framework) evolves. This mission formalizes that online rounding scheme's three defining probabilistic guarantees and the resulting competitive-ratio theorem.

Setting

Fix a rooted tree T = (V, E, r) with non-negative edge costs c : E → ℝ, and k groups g₁, …, g_k ⊆ V, each request (r, gᵢ) arriving online. An online covering algorithm (from Chapter 4's framework, applied to the LP relaxation of this connectivity problem) maintains a monotonically increasing fractional weight w : E → ℝ on the edges, reinterpreted so that wₑ is the maximum flow that can be routed through e to any vertex of its subtree — a technical substitution needed so weights are monotone non-increasing along any root-to-leaf path, the property the rounding algorithm requires. At the end of each iteration in which some weights are augmented from w to w' = w + δ, the rounding algorithm processes every edge e with δₑ > 0, in topological order starting from the root, and randomly decides whether to add it to a growing random edge-cover C ⊆ E: deterministically, if w'ₑ > 1; via a single coin flip, if e is incident to the root or its parent edge's inclusion in C is already certain; via a coin flip conditional on the parent edge already being in C, otherwise. Because a coin is only ever flipped for a child once its parent is (or is already known to be) in C, C always induces a connected subtree containing the root.

Formalization targets

Theorem 11.4 (the goal, p. 231): there is a randomized online algorithm for the group Steiner problem in trees with competitive ratio O(log²n log k), where n is the number of leaves. It is built by running T independent trials of the rounding scheme in parallel and taking the union of the resulting covers, for T chosen (this mission's own explicit derivation — the book gives only the narrative "we run O(log k log N) independent trials... using simple probabilistic analysis") so that every group fails to be covered with probability at most 1/(2k), while the union's expected cost stays at T · log(n) · OPT.

Three milestones, in attack order, each stated exactly as the book states it (p. 230-231), with the book's own caveat "we state the main lemmas and omit the proofs" preserved — no in-source proof exists for any of the three beyond the algorithm's own description, so each is left sorry with no invented proof strategy:

Lemma 11.1: at the end of an iteration, ℙ[e ∈ C] = w'ₑ for every edge, and ℙ[e ∈ C] = 1 whenever wₑ > 1 already.
Lemma 11.2: the expected cost of C is at most ∑_{e∈T} cₑ w'ₑ (linearity of expectation applied to Lemma 11.1).
Lemma 11.3: for a group g of size at most N with total routable flow wg ≥ 1, the probability some vertex of g is covered is Ω(1/log N).

Significance

This is the survey's most involved application of the primal-dual framework: unlike Chapters 5, 9, 10 and 13, which round a single scalar decision per online step, the group Steiner algorithm must couple an entire iteration's worth of edge decisions so that the resulting random set stays a connected subtree — the coin-flip probabilities in the Algorithm box are exactly the minimal adjustment needed to keep marginal probabilities matching the fractional solution while preserving this connectivity invariant online. No formal development of the group Steiner problem (online or offline) was found on the platform as of 2026-09-20; this mission is the first.

Difficulty

Two distinct obstacles. First, faithfully representing "the probability that e ∈ C" for an online, coupled random process without assuming its proof: the mission represents the algorithm's random cover as an abstract finite probability distribution RandomCover E and states each lemma as an implication from the Algorithm box's three coupling rules (transcribed as hypotheses on marginal and conditional probabilities) to the claimed marginal or expected-value conclusion — capturing exactly what the book asserts without proof, rather than either assuming the conclusion trivially or constructing a full multi-iteration coupled process (which the source's own "we omit the proofs" indicates is genuinely nontrivial, citing [56]). Second, Theorem 11.4's own competitive ratio is stated in the book only asymptotically, with a purely narrative derivation ("we run O(log k log N) independent trials... we get a competitive ratio of O(log n log k log N)... probability at least 1 − 1/k") and no displayed formula anywhere in the chapter. Per this series' explicit-constants rule, this mission supplies its own explicit closed form for the number of trials T and the resulting bounds via a standard Chernoff/union-bound argument applied to Lemma 11.3's constant α; this derivation is the mission's own (documented below), not a transcription, since none exists in the source to transcribe.

Formalization scope

RandomCover E is a finite pmf p : Finset E → ℝ (Finset E itself finite since E is Fintype), with marg, condProb and expectedCost/probHits derived from it by ordinary Finset sums — no measure theory, since the sample space is always finite. RoundedTree E bundles parent : E → Option E (e(p)) and a non-negative cost. Lemma 11.3's wg (the flow routable to a group's vertices simultaneously) is left as hypothesis-supplied data rather than defined via an explicit max-flow formalization, which this mission's scope does not require (welcome contribution). Theorem 11.4's number of trials is the explicit closed form T = ⌈(log N · log(2k)) / α⌉, α the (existentially quantified, uniform) constant from Lemma 11.3; its coverage guarantee is 1 − 1/(2k) per group (this mission's own union-bound derivation), not the book's stated 1 − 1/k — the book reaches the stronger bound via an additional shortest-path fallback mechanism for any group the trials miss, which is out of scope here (welcome contribution, along with completing any of the four sorrys and formalizing Theorem 11.5's extension to general graphs via HST embedding, out of scope since it depends on an external embedding result not proved in this book).

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024
N. Garg, G. Konjevod, R. Ravi. A polylogarithmic approximation algorithm for the group Steiner tree problem. Journal of Algorithms, 37(1):66-84, 2000 (cited as [56] in the survey).
N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor. A general approach to online network optimization problems. ACM Transactions on Algorithms, 2(4):640-660, 2006 (cited as [4], the source of this chapter's results per the Notes, p. 231).

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Algorithmic Game Theory Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach VI: Maximizing Ad-Auctions RevenueTextbook

Motivation

Search-engine ad-auctions sell items (ad slots) to buyers who arrive with per-item bids and a fixed daily budget, online: each item must be allocated the moment it appears, with no ability to revisit past allocations, and no buyer may ever be charged more than its budget. Buchbinder and Naor's survey [1] models this as a generalization of online bipartite matching and derives an allocation algorithm through the same online primal-dual recipe formalized elsewhere in this series (Chapter 4's framework), applied to a genuinely different LP shape — a maximization (revenue) problem whose "packing" role and "covering" role are the reverse of Chapters 4, 5 and 7 — and to a different constraint structure (budget caps on a per-buyer accumulated sum, not a per-constraint 0/1 covering requirement). This mission covers Section 10.1, "The Basic Algorithm": the single-slot allocation algorithm and its $(1-1/c)(1-R_{\max})$ -competitive analysis (Theorem 10.1). Sections 10.2-10.3 (multiple ad-slots via strong duality; stochastic per-buyer spending guarantees) are out of scope — see Formalization scope.

Setting

Fix a finite set $I$ of buyers, each with budget $B_i > 0$ , and a finite set $M$ of items; buyer $i$ bids $b_{ij} \ge 0$ for item $j$ , revealed one item at a time in the order enumerated by $M$ . Let $R_{\max} := \max_{i,j} b_{ij}/B_i$ , carried as an explicit positive parameter. The Allocation algorithm (p. 212), upon each item $j$ 's arrival, allocates it to the buyer $i$ maximizing $b_{ij}(1-x_i)$ (where $x_i \in [0,1]$ is buyer $i$ 's current primal value); if $x_i \ge 1$ already, nothing happens (the buyer is "full"). Otherwise it charges buyer $i$ the minimum of $b_{ij}$ and its remaining budget, sets the dual allocation variable $y_{ij} \leftarrow 1$ , sets $z_j \leftarrow b_{ij}(1-x_i)$ , and updates $x_i \leftarrow x_i(1+b_{ij}/B_i) + b_{ij}/((c-1)B_i)$ for a constant $c$ fixed by the analysis. The revenue actually collected from buyer $i$ is $\min\!\big(\sum_j b_{ij}y_{ij},\,B_i\big)$ — the buyer is never charged more than its budget. Unlike Chapters 4/7, this LP's dual (the ad-auctions revenue objective, Fig. 10.1) is the maximization problem being solved online; the "primal" covering LP ( $x_i$ , $z_j$ variables) exists only as a duality certificate.

Formalization targets

Theorem 10.1 (the goal, p. 212), given the milestone's dual near-feasibility bound and the fact that each buyer's total accrued bids exceed its budget by at most a factor of $R_{\max}$ (Claim (3)'s consequence):

\forall\, (x'', z'')\text{ feasible for Fig. 10.1's covering LP},\ \ \textstyle\sum_i \mathrm{actualCharge}(i) \ \ge\ (1-\tfrac1c)(1-R_{\max}) \Big(\textstyle\sum_i B_i x''_i + \sum_j z''_j\Big),

with $c = (1+R_{\max})^{1/R_{\max}}$ taken verbatim from the theorem's own statement — the exact formula, not an $O(\cdot)$ instantiation. Inequality (10.1) (p. 213), the milestone, is the book's own induction-proved lower bound on a buyer's primal value in terms of its accrued bids: $x_i \ge \frac{1}{c-1}\big(c^{(\sum_j b_{ij}y_{ij})/B_i} - 1\big)$ .

Significance

Theorem 10.1 is the entry point to a short but influential sub-line of the online primal-dual method — Section 10.2 extends it to multiple ad-slots via strong duality for maximum-weight matching (rather than the weak duality this framework otherwise relies on throughout), and Section 10.3 incorporates stochastic per-buyer spending guarantees, both reusing this section's constant $c=(1+R_{\max})^{1/R_{\max}}$ and its limit $c\to e$ as $R_{\max}\to0$ (recovering the classic $(1-1/e)$ -competitive ratio for the unweighted, unbudgeted case). It is also the first mission in this series to apply the online primal-dual method to a genuine revenue-maximization problem rather than a covering/packing pair with matching roles. No formal development of ad-auctions, budgeted online matching, or this constant was found on the platform as of 2026-09-20 (searches below); this mission is the first.

Difficulty

As with 04-framework's Algorithm 3 and 07-generalized-caching's Fractional Caching algorithm, the central obstacle is characterizing an online process by its own final output. Unlike those two chapters, however, step (3)'s update increment varies per allocation (bd, the specific bid of the item just won), so no closed-form solution of the recurrence exists in general; buyerX is instead defined as an explicit List.foldl realizing the exact per-step update, over the (temporally ordered) list of bids a buyer actually won — a faithful, if less immediately readable, transcription of "the algorithm's output as a function of its own trajectory," in the same spirit as this series' other closed-form definitions. A second difficulty specific to this chapter: the book's derivation of inequality (10.1) is itself an induction on iterations (not a single algebraic step, unlike Eq. (7.2) in 07-generalized-caching), and the theorem's final bound further combines it with a separate "at most one undercharged iteration" argument (Claim (3)'s conclusion, p. 214-215) turning the raw accrued-bid bound into one about the actually collected (budget-capped) revenue — both are stated here as explicit hypotheses (the milestone, and h_at_most_one_undercharge) rather than derived, since the goal is a faithful statement, not a proof; both sorrys are documented, not silently discharged.

Formalization scope

AdAuctionsInstance I M bundles b : I → M → ℝ, B : I → ℝ (hB_pos), and Rmax : ℝ with hRmax_pos : 0 < Rmax and hRmax_bound : ∀ i j, b i j ≤ Rmax * B i — the last two as explicit hypotheses, never derived via Finset.sup, matching 04-framework's d and 07-generalized-caching's k. cParam inst := (1+Rmax)^(1/Rmax) uses Real.rpow (ℝ^ℝ). buyerX inst i bids folds step (3)'s update over a list of won bids; revenue/actualCharge realize $\sum_j b_{ij}y_{ij}$ and its budget-capped charge. Reals throughout. Explicitly out of scope: Section 10.2's multiple-slot generalization (Theorem 10.2), which requires strong duality for maximum-weight bipartite matching as an explicit premise (the book: "our analysis... crucially relies on strong duality") — a substantially different LP structure (an integral matching LP, not this section's per-buyer budget LP) that this mission's AdAuctionsInstance does not model, and whose applicability of PrimalDualOnline.LP.strong_duality_adapter (this book's own Theorem 2.2) was not verified in the time available. Section 10.3's stochastic guarantee (Theorem 10.3) is likewise out of scope, and its own BRIEF.md-flagged ambiguity (whether its scalar g is $\min_i g_i$ or another aggregate of the per-buyer vector $g_i$ ) was not resolved. Both are natural follow-on missions, not attempted here. Welcome contributions: completing the two sorrys, and the Section 10.2-10.3 follow-on mission(s).

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024

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The Design of Competitive Online Algorithms via a Primal-Dual Approach V: Online RoutingTextbook

Motivation

Network routing is one of the paradigmatic applications of the online primal-dual method: requests for bandwidth between a source and target arrive one at a time, and the algorithm must commit bandwidth to paths without knowing future requests. Chapter 4 already gave a simple (3, O(log n))-competitive routing algorithm as an illustration of the general framework. This chapter asks for something qualitatively stronger: a uni-criteria (1, O(log n))-competitive algorithm — one that routes the full optimal bandwidth (no loss on the throughput side at all), paying only in a bounded edge-capacity violation. The chapter shows this exact-throughput guarantee is achievable, and is moreover the key building block for several other routing objectives (fair routing, max-min fairness) built on top of it in Section 9.2 — a (1, O(log n))-competitive algorithm composes into those richer objectives in a way a merely constant-factor-lossy algorithm does not.

Setting

Fix a graph $G=(V,E)$ , $|V|=n$ , $|E|=m$ , with integer edge capacities $u: E \to \mathbb{N}$ . Routing requests $r_i$ arrive online, each demanding one unit of bandwidth between a source and target; in the splittable model (this chapter's comparison class), a request's bandwidth may be divided across multiple paths. A $(c_1,c_2)$ -competitive algorithm routes at least $1/c_1$ of the maximum possible bandwidth while guaranteeing every edge's load (bandwidth allocated divided by capacity) is at most $c_2$ . The chapter's generic algorithm (Section 9.1) maintains, for each of $O(\log n)$ copies $G_0,\dots,G_k$ of the graph (copy $j$ keeping only edges of capacity at least $m^j$ , each capped at $\min(u(e), m^{j+2})$ ), a primal-dual pair matching Chapter 4's own routing LP (Fig. 9.1, identical to Fig. 4.2): a request is routed on the shortest path (by the copy's current primal edge-lengths $x(e,j)$ ) if that length is below $1$ , multiplicatively updating $x(e,j)$ on the path's edges; otherwise, subject to a capacity-limited fallback rule, on an arbitrary feasible path.

Formalization targets

Theorem 9.2 (goal, p. 204): the algorithm is (1, O(log n))-competitive with respect to all splittable routing solutions — exact throughput (ratio 1), edge load at most O(log n).

Lemma 9.1 (milestone, p. 202-204): a single copy $G_j$ 's own guarantee, which Theorem 9.2's proof composes across all copies: the algorithm accepts at least $M$ (the maximum splittable bandwidth achievable in $G_j$ , out of the requests introduced to it) and incurs load $O(\log n)$ on every edge of $G_j$ .

Significance

This is the survey's demonstration that the online primal-dual method, in its most basic form (a single accumulating dual sum driving a multiplicative primal update, exactly Chapter 4's framework), scales to a genuinely harder bicriterion objective once composed across a carefully constructed family of graph copies — the copies are what let the algorithm avoid ever needing to reason about which of the exponentially many $s_i$ - $t_i$ paths to consider, reducing routing to a sequence of independent shortest-path computations. The chapter's own Section 9.2 builds a coordinate-wise-competitive fair-routing algorithm directly on top of Theorem 9.2 (not formalized here), and its Notes section places $(1, O(\log n))$ -competitiveness as "a crucial non-trivial step" the chapter needed before those richer objectives became tractable at all. No prior formalization of online routing, splittable or otherwise, was found on the platform as of 2026-09-20 (search below).

Difficulty

This is the most algorithmically intricate chapter in this series: the algorithm processes each request against every one of $O(\log n)$ graph copies, each copy running its own instance of the Chapter-4-style primal-dual update, and Theorem 9.2's own proof composes Lemma 9.1's per-copy guarantee via a combinatorial backward induction (partitioning the offline-optimal solution's paths into groups by bottleneck capacity, and showing group by group that the algorithm's cumulative routed bandwidth across the top levels dominates the cumulative optimal bandwidth in those groups) together with a separate geometric argument bounding how many copies any single edge can meaningfully appear in. Fully modeling the copy construction, the request-routing process, and both composition arguments from first principles was judged to exceed this mission's time budget without sacrificing the faithfulness of what does get stated (per CAPTAIN_BRIEF.md rule 6). Instead: Lemma 9.1 is formalized via the two facts its own proof isolates as doing the real work — a weak-duality contradiction bound (stepB ≥ M − uMin) and the step-(1c) fallback's own greedy-fill rule for part (i); the multiplicative-update invariant x(e,j) ≤ 2 for part (ii), whose consequence — the exact constant 2 + 6·log₂n — is re-derived from scratch in this mission (the displayed equation this derivation depends on was garbled by the PDF's text extraction; it was confirmed against the actual typeset page image before drafting, see SELF_REVIEW.md). Theorem 9.2 then composes Lemma 9.1 across copies via two explicit, clearly-labeled hypotheses standing for the book's own backward-induction accounting and edge-multiplicity argument, respectively — genuine mathematical content this mission does not re-derive, named honestly as hypotheses rather than silently assumed away or approximated by a weaker statement.

Formalization scope

No shared data structure was introduced: every quantity in both theorems (bandwidths, capacities, loads) is a plain real-number hypothesis-level parameter, since neither theorem's own content needs a reusable instance record (unlike the packing/covering CoveringInstance of 04-framework, this chapter's per-copy quantities are consumed once each, not threaded through a family of algorithms). per_copy_guarantee (Lemma 9.1) takes the weak-duality bound, the step-(1c) fill rule, and the x(e,j)≤2 invariant as hypotheses and derives both parts of the lemma's conclusion by real algebra (a sign case-split for part (i); Real.logb/rpow manipulation for part (ii)). routing_competitive (Theorem 9.2) takes Lemma 9.1's guarantee (universally quantified over the copy index J, a general Fintype) plus the two composition hypotheses described above, and derives the bicriterion conclusion by summation, transitivity, and scaling. This correctly rules out the trivializing formalization in which the composition hypotheses are strengthened to directly assert the theorem's own conclusion (each is a strictly weaker, independently-motivated fact — the backward-induction accounting identity and the geometric edge-multiplicity bound — checked in MODERATION_NOTES.md against this exact failure mode). Reals throughout; Real.logb 2 for log₂. Welcome contributions: completing the two sorrys (Lemma 9.1's part (i) is short algebra; part (ii) needs Real.rpow/Real.logb lemmas; Theorem 9.2's is transitivity/summation once its hypotheses are in hand), and — the natural follow-on — formalizing the copy construction and the backward-induction/edge-multiplicity arguments hquota/hload_aggregation currently stand in for, which would upgrade them from hypotheses to theorems in their own right; Section 9.2's coordinate-wise-competitive fair-routing algorithm (Theorem 9.3) and the matching lower bound (Lemma 9.5) are further natural follow-ons.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024

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The Design of Competitive Online Algorithms via a Primal-Dual Approach IV: Generalized CachingTextbook

Motivation

Caching is a two-level memory-management problem — the fast level (cache) can hold only $k$ items, and the algorithm must decide, online, which item to evict whenever the current request misses — that is normally analyzed through the competitive ratio of ad hoc marking or LRU-style rules. Buchbinder and Naor's survey [1] instead recasts weighted caching (non-uniform fetching costs) as an instance of the covering/packing linear program, and derives a fractional online algorithm through the same primal-dual recipe formalized in this series' 04-framework mission (Chapter 4), but for a genuinely different LP shape: the caching LP's right-hand side varies from constraint to constraint, unlike Chapter 4's uniform $b(j)=1$ . This mission covers Sections 7.1-7.2 of Chapter 7, "Generalized Caching": the fractional weighted-caching algorithm and its $2(1+\ln k)$ -competitive analysis. Sections 7.3-7.4, which further generalize to non-uniform page sizes (not just costs), are out of scope — a natural follow-on mission, not attempted here (see Formalization scope).

Setting

Fix a finite set $V$ of primal variables $x(p,j)$ — one per page $p$ and each of its eviction intervals between its $j$ -th and $(j{+}1)$ -th request — with fetching cost $c(p,j) = c_p \ge 1$ (the book's standing weighted-caching assumption), and a finite set $\mathrm{Time}$ of online constraints, one per request time $t$ , revealed in the order enumerated by $\mathrm{Time}$ . The eviction-charged LP formulation (the book charges for evicting pages rather than fetching them, an equivalent reformulation up to an additive constant independent of the request sequence) constrains, at each time $t$ : $\sum_{v \in S(t)} x_v \ge \mathrm{rhs}(t)$ , where $S(t)$ is the set of currently-active eviction variables for pages present until $t$ (excluding the page just requested) and $\mathrm{rhs}(t) = |B(t)| - k$ is the amount of cache space those pages must collectively vacate. The Lagrangian dual has a variable $y(t)$ per request time and a variable $z(p,j)$ per eviction interval, with dual constraint $\big(\sum_{t \mid v \in S(t)} y(t)\big) - z_v \le c_v$ . As in Chapter 4, primal variables may only increase and the algorithm sees each constraint only upon its arrival.

The Fractional Caching algorithm (p. 153-154) sets each $x(p,j)$ to jump from $0$ to $1/k$ the first time its dual constraint tightens, then increases continuously according to an exponential function of the accumulated dual sum until it saturates at $1$ (at which point $z(p,j)$ begins absorbing further dual increase at the same rate, freezing $x(p,j)$ ). This is a genuinely different LP shape from Chapter 4's framework (non-uniform, time-varying right-hand side) reusing the same complementary-slackness design pattern as that chapter's Algorithm 3.

Formalization targets

Theorem 7.1 (the goal, p. 154), given the algorithm's final dual values $y \ge 0$ , $z \ge 0$ and primal feasibility:

\Big(\forall v,\ \textstyle\sum_{t \mid v \in S(t)} y_t - z_v \le c_v(1+\ln k)\Big) \ \Longrightarrow\ \Big(\forall x''\text{ feasible},\ \textstyle\sum_v c_v x_v \le 2(1+\ln k)\sum_v c_v x''_v\Big),

i.e. the algorithm is $2(1+\ln k)$ -competitive, with the constant taken verbatim from the book's own theorem statement (no $O(\cdot)$ instantiation needed here, unlike most goals in this series). The antecedent is itself Eq. (7.2) (p. 155), formalized as the milestone dual_near_feasible: the algorithm's dual solution, scaled down by $1+\ln k$ , is feasible — the book's own intermediate step, derived from the fact that every cachingX value is capped at $1$ .

Significance

Chapter 7 is the first chapter in this survey to apply the online primal-dual method to an LP whose right-hand side is not uniformly $1$ (unlike Chapters 4 and 5), demonstrating the method's reach beyond the "simplified" 0/1-coefficient covering LP that 04-framework formalizes. The $2(1+\ln k)$ fractional guarantee is also the analytical core of the chapter's randomized rounding result (Theorem 7.3, not part of this mission — see Formalization scope), which converts it into an actual $O(\log k)$ -competitive randomized algorithm against an adaptive adversary, and of the chapter's further generalization to non-uniform page sizes (Theorem 7.5, Sections 7.3-7.4). No formal development of weighted or generalized caching was found on the platform as of 2026-09-20 (searches below); the existing KServer.* namespace formalizes a different, unweighted, uniform $k$ -server model and shares no substrate with this mission. This mission is the first.

Difficulty

As with 04-framework's Algorithm 3, the central obstacle is characterizing an online process by its final output alone: cachingX is defined as the algorithm's own closed-form update rule (threshold-then-exponential, capped once $x(p,j)=1$ ), evaluated at the run's final accumulated dual values, rather than as an independently-constrained free variable — the latter would let $x$ and $y$ be chosen to satisfy the conclusion's inequalities directly, trivializing the claim that a specific online algorithm achieves this ratio. Establishing that the capped closed form is faithful (not merely an invented convention) requires the same monotonicity argument 04-framework's alg3X uses, adapted to this chapter's extra $z(p,j)$ term inside the exponent (present here; absent from Chapter 4's Algorithm 3). The proof's own structure — splitting the primal cost into a $0\to1/k$ contribution ( $C_1$ ) bounded via complementary slackness and a $1/k\to1$ contribution ( $C_2$ ) bounded via a derivative/telescoping argument over the continuous accumulation process (Eqs. (7.6)-(7.10), p. 155-157) — is, as in Chapter 4, a genuinely dynamic fact about the trajectory, not encoded as a hypothesis; the mission states the theorem faithfully and leaves the sorry for that argument, per this series' documented-simplification convention.

Formalization scope

CachingInstance V Time bundles S : Time → Finset V, rhs : Time → ℝ (unlike 04-framework's CoveringInstance, whose right-hand side is fixed at $1$ throughout), c : V → ℝ with hc_pos : ∀v, 1 ≤ c v (the book's own literal cp ≥ 1, not a strengthening), and k : ℕ with hk_pos : 0 < k. dualSum inst y v := ∑_{t \mid v \in S(t)} y_t, matching 04-framework's pattern. cachingX inst y z v is a noncomputable def: 0 before activation, otherwise min 1 ((1/k) exp((dualSum - z - c v)/c v)), so "the algorithm's output" is genuinely a function of its dual trajectory. Reals throughout; Real.log for the book's natural log $\ln$ . Explicitly out of scope: Section 7.3's rounding apparatus (Theorem 7.3, the map from fractional to randomized-integral cache states) and Section 7.4's non-uniform-page-size generalization (Theorem 7.5) — both are natural follow-on missions building on this one's CachingInstance and cachingX, not attempted here per this chunk's own BRIEF.md, which flags Theorem 7.1 alone as "a complete, self-contained mission goal" when the rounding apparatus proves too heavy for a single pass. Welcome contributions: completing the two sorrys, and the Section 7.3-7.4 follow-on mission.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024

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Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach III: Metrical Task Systems on a Weighted StarTextbook

Motivation

The metrical task system (MTS) problem is one of the earliest and most general online models: a server occupies a state in a metric space, requests arrive with per-state service costs, and the server may change state (paying the metric's transition cost) before serving each request. On a general metric, competitive algorithms are hard to design directly. Chapter 6 shows that on a weighted star metric — the simplest genuinely non-uniform metric, a hub with leaves at varying distances — the online primal-dual framework of Chapter 4 becomes applicable, but only after a change of rules: the chapter defines a new MTS model, in which the server may change state only at the boundary of a "phase" (an interval during which its accumulated service cost reaches the state's own transition charge), and shows this new model is cost-equivalent, up to a constant factor, to the standard model. This equivalence is what licenses recasting the (new-model) problem as a covering linear program with the online primal-dual framework directly applicable — the chapter's actual algorithmic payoff (an unnumbered $O(\log N)$ -competitive result, Section 6.2) rests entirely on it.

Setting

Fix a set of leaves $V$ (the book's finite $\{1,\dots,N\}$ ) of a weighted star, each leaf $i$ at distance $d'(i) \ge 0$ from the center. The chapter immediately collapses the full star metric to a single per-state transition charge $d(i) := 2d'(i)$ , since on a star every transition $i \to j$ costs at most $d'(i) + d'(j)$ , and charging the doubled source leaf's distance alone (never charging for arriving) upper-bounds every transition cost independent of destination. A standard-model solution is a finite sequence of runs, each specifying a state $s_i$ occupied and the service cost $w_i \ge 0$ accumulated while in that state, ending with a transition (cost $d(s_i)$ ) to the next run's state; its total cost is $\sum_i (w_i + d(s_i))$ . A new-model solution is a finite sequence of phases, each specifying a state visited; a phase's cost is exactly $d(\text{state})$ regardless of how much service actually occurred during it (a state change is only permitted once a phase's accumulated service reaches the phase's own $d$ -value), so the new model's total cost is $\sum_i d(s_i)$ .

Formalization targets

Lemma 6.1 (goal, p. 144): any standard-model solution's runs $(s, w)$ transform into a new-model solution whose cost is at most $2 \cdot \sum_i (w_i + d(s_i))$ ; in particular $OPT_n(\bar\sigma) \le 2 \cdot OPT_o(\bar\sigma)$ .

Lemma 6.2 (milestone, p. 145): any new-model solution's phases $(s, w)$ , with each phase's actual service $w_i \le d(s_i)$ , are simultaneously a legal standard-model solution (the same trajectory, recosted) whose standard-model cost is at most $2 \cdot \sum_i d(s_i)$ .

Together (stated in the book but not separately numbered, hence not a formalization target here): a $c$ -competitive algorithm in the new model implies a $4c$ -competitive algorithm in the standard model.

Significance

This is a model-equivalence result, a distinct and recurring pattern in online algorithm design from the competitive-ratio bounds formalized elsewhere in this series: rather than analyzing an algorithm directly, the chapter first shows that solving an easier, more restricted version of the problem (state changes only at phase boundaries) loses only a constant factor, and only then designs an algorithm for the restricted version. The technique generalizes (the chapter's own Notes section places it in the context of Borodin et al.'s original MTS bounds, and of later work on hierarchically well-separated trees for general metrics), but this chapter gives its cleanest, self-contained instance: two short, tight (factor-2 each direction) transformations between two formally distinct online cost models. No prior formalization of metrical task systems on a weighted star, or of this new/standard model equivalence, was found on the platform as of 2026-09-20 (search below); the platform's existing KServer.* campaign formalizes a different online problem (uniform-metric $k$ -server) with a different metric structure and is not adjacent substrate for this chapter's weighted-star MTS model.

Difficulty

The central obstacle is representing "a solution" at a level of abstraction faithful to the book's own proof without committing to a full continuous-time process model (states as functions of a real time variable, phases as recursively-defined stopping times, requests as an explicit arriving sequence) that neither lemma's own proof actually needs. Both proofs work entirely at the granularity of a solution's runs (standard model) or phases (new model) — finite sequences of (state, cost) data — never referencing continuous time except to justify that this decomposition exists. This mission formalizes both lemmas at exactly that granularity: a run/phase sequence indexed by Fin k, with costStandard/costNewPhases the book's own displayed cost formulas. Lemma 6.1's proof genuinely constructs a new object (the delayed-transition solution $S'$ ) and only bounds its cost, never gives $S'$ a closed form — formalized here as an existential over a per-segment cost witness w', bounded above and below exactly as the proof's own argument does (with the lower bound d(s i) ≤ w' i serving as the guard against the vacuous witness w' = 0, since without it the existential is trivially satisfiable and asserts nothing). Lemma 6.2's proof, by contrast, reuses the same trajectory in both models with no construction at all, formalized directly as a cost comparison between costStandard and costNewPhases applied to the identical (s, w) data.

Formalization scope

WeightedStar V bundles centerDist : V → ℝ (the book's $d'$ ) with non-negativity; WeightedStar.d is the collapsed charge $d(i) = 2d'(i)$ . costStandard/costNewPhases are literal transcriptions of the two models' displayed cost formulas over a Fin k-indexed run/phase sequence. standard_to_new (Lemma 6.1) is the existential described above; new_to_standard (Lemma 6.2) is the direct cost comparison. Reals throughout; V is left a general Type* (not assumed Fintype) since neither lemma's own statement needs the chapter's finiteness assumption $|V|=N$ — that assumption only matters for the unnumbered $O(\log N)$ -competitive claim of Section 6.2, out of scope per BRIEF.md's explicit instruction (no numbered theorem to formalize it against). This correctly rules out the trivializing formalization in which "new-model solution" is left as an unconstrained free variable satisfying only the conclusion's own inequality, or in which the star structure is dropped entirely in favor of an arbitrary metric (the chapter's own reduction to a per-state charge $d(i)$ , rather than a full metric $d: V\times V\to\mathbb R$ , is precisely what the star's structure licenses, and is preserved here via WeightedStar.d rather than a bare hypothesis-level function). Welcome contributions: completing the two sorrys (Lemma 6.1's needs an explicit construction of $S'$ and its per-segment cost bound; Lemma 6.2's is a short termwise algebraic argument), and formalizing the Section 6.2 covering-LP algorithm and its $O(\log N)$ -competitive claim as a follow-on mission once it can be stated against a numbered result.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024
Bansal et al., reference [14] of this chapter's own bibliography (not independently verified by this mission), cited by the book as the source of the weighted-star MTS results this chapter presents ("The results in this chapter are based on the work of Bansal et al. [14]," p. 147).
Borodin, Linial, Saks, reference [29] of this chapter's own bibliography (not independently verified by this mission), cited as the paper that originally formulated the standard MTS model and its tight $2N-1$ deterministic bound (p. 147).

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Linear Optimization Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach II: The Online Set-Cover ProblemTextbook

Motivation

Section 4 of this survey derives a simple randomized $O(\log m \log n)$ -competitive algorithm for the online set-cover problem, by rounding the fractional solution the online packing-covering framework produces. An intriguing question the survey poses next: can the same guarantee be achieved deterministically? The standard tool for removing randomness, the method of conditional expectations, requires finding a pessimistic estimator — a potential function whose value the algorithm can track and whose behavior certifies the randomized algorithm's guarantee step by step. Chapter 5 constructs exactly this potential function for the weighted online set-cover problem, and shows that greedily minimizing it online reproduces the randomized algorithm's competitive ratio with no randomness at all. This mission formalizes that construction: the potential function itself (Lemma 5.1) and the correctness guarantee it buys (Theorem 5.2).

Setting

Fix a finite universe of elements $E$ and a finite family of sets $T$ with positive costs $c_s$ , both known to the algorithm in advance (only which elements will actually need covering, and in what order, is unknown). A monotonically increasing assignment $w : T \to \mathbb{R}$ of fractional weights to sets is produced online by a fractional subroutine (any $O(\log m)$ - competitive online fractional algorithm — the survey's own Section 4.2 supplies one). An element's weight is $w_e := \sum_{s \mid e \in s} w_s$ . Given a target $\alpha \ge c(C_{OPT})$ (a guessed upper bound on the optimal integral cover's cost — the survey handles an unknown optimum by doubling this guess across phases, outside this chapter's own scope), the algorithm maintains a chosen cover $C \subseteq T$ and the potential

\Phi \;=\; \sum_{e \notin \bar C} n^{2w_e} \;+\; n \cdot \exp\!\Big(\tfrac{1}{2\alpha} \sum_{s} \big(c_s \chi_C(s) - 3 w_s c_s \log n\big)\Big),

where $n = |E|$ and $\chi_C$ is $C$ 's characteristic function. Whenever a set $s$ 's weight increases, the algorithm computes $\Phi$ both with and without adding $s$ to $C$ and chooses whichever keeps $\Phi$ from exceeding its value before the step (failing only if neither does, which Lemma 5.1 shows cannot happen when $\alpha \ge c(C_{OPT})$ ).

Formalization targets

Theorem 5.2 (goal, p. 139): given the invariant $\Phi < n^2$ that Lemma 5.1 maintains throughout a run, (i) every element of weight $\ge 1$ is covered, and (ii) the chosen cover costs at most $\alpha \cdot O(\log m \log n)$ .

Lemma 5.1 (milestone, p. 137): the potential function never increases in expectation across a weight-augmenting step, under the algorithm's own randomized choice of whether to add the augmented set to the cover (the internal argument — via the method of conditional expectations — that certifies the deterministic algorithm's choice rule never fails).

Significance

This chapter answers, for the online set-cover problem specifically, a question that recurs throughout online algorithm design: when can a randomized guarantee be derandomized online? The potential-function technique here is the survey's own template for the answer (it recurs, per the chapter's Notes section, in the routing algorithm of Chapter 9 and the ad-auctions algorithm of Chapter 10, both formalized as separate missions in this series) — a self-contained, reusable instance of "derandomization via an explicit pessimistic estimator" in the online setting, distinct from the offline set-cover primal-dual and dual-fitting algorithms of this book's own Chapter 2, already on the platform (PrimalDualOnline.SetCover.*, checked below: a static instance with no arrival order and no potential function, a genuinely different model).

Difficulty

The central formalization challenge is that Lemma 5.1's own statement, " $\Phi_{end} \le \Phi_{start}$ ", denotes the potential's value in expectation under the algorithm's randomized choice — not a single deterministic before/after pair — since the lemma is proved via a probabilistic argument (adding $s$ to the cover with probability $1 - n^{-2\delta_s}$ ) whose role is purely internal to justifying the deterministic algorithm's rule (choose whichever of the two options controls $\Phi$ ). Stating the lemma as a bare inequality between two potential values, without the mixture, would either be false (the "add $s$ " branch alone can increase $\Phi$ ) or would silently smuggle in the derandomized choice as a hypothesis rather than proving it is always available. This mission states the expectation explicitly as a probability-weighted average of the two branch potentials, matching the actual analytic content of the book's proof (equations 5.1-5.6) rather than its final one-line restatement.

Formalization scope

SetCoverInstance E T bundles elemSets : E → Finset T (the sets containing an element) and positive costs c. elementWeight and coveredBy are literal transcriptions of $w_e$ and " $e \in \bar C$ ". potential transcribes $\Phi$ 's displayed formula verbatim, with n cast from Fintype.card E. potential_nonincreasing (Lemma 5.1) is the expectation inequality described above. algorithm_correctness (Theorem 5.2) takes the potential invariant Φ < n² as a hypothesis (the state Lemma 5.1, applied repeatedly from the initial value $\Phi < n^2$ , is what the book's own proof shows every reachable state satisfies) together with an explicit ratio β standing for "the fractional solution is O(log m)-competitive" (∑ wₛcₛ ≤ βα) — the book imports this fact from Section 4 as a black-box subroutine rather than re-deriving a specific numeric constant in this chapter, and this mission does the same rather than re-deriving Chapter 4's own constant under the (different) $d \to m$ substitution the book's prose glosses over. The conclusion is then the fully explicit α · log n · (3β + 2), matching the book's own derivation (displayed inequality, p. 139-140) with O(log m) replaced by the parameter β. This correctly rules out the trivializing formalization in which the O(log m log n) bound is left as an unquantified existential constant, or in which Φ's invariant is assumed directly as an unmotivated free hypothesis rather than the fact Lemma 5.1 is what actually establishes. Reals throughout; Real.log, Real.exp, Real.rpow (via the ^ notation on reals) for the book's own log, exp and n^{2w_e}. Nothing here is reused from 04-framework (concurrent draft; per this series' own rule, drafts do not import drafts) even though this chapter's fractional subroutine is conceptually the same online covering framework — restated here only as the abstract ratio β, not as a Lean dependency. Welcome contributions: completing the two sorrys, and formalizing the doubling-across-phases wrapper (Section 5.1's "Obtaining a Deterministic Algorithm" discussion) that removes the need to know α ≥ c(C_OPT) in advance.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024
N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor. The online set cover problem. STOC 2003 / SIAM J. Comput. 39(2), 2009 (cited by this book's Chapter 5 Notes as [3]).

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Linear Optimization Operations Research·Captain: mikedeng1

The Design of Competitive Online Algorithms via a Primal-Dual Approach I: The Online Packing-Covering FrameworkTextbook

Motivation

Many online problems — renting vs. buying equipment, routing traffic without knowing future demand, allocating advertising budget as bids arrive — share a common linear-programming shape: a covering (minimization) problem whose constraints appear one at a time, or its dual packing (maximization) problem whose variables appear one at a time, with no ability to revisit past decisions. Buchbinder and Naor's survey [1] recasts the online primal-dual method, originally developed for offline approximation in Section 2 of the same survey (already covered on this platform via the PrimalDualOnline namespace, citing Buchbinder's thesis [2]), as a general recipe for such online problems, unifying earlier ad hoc analyses of the ski-rental problem (Chapter 3) and paving the way for the online set-cover, routing, caching, and ad-auction algorithms formalized elsewhere in this series. This mission covers Chapter 4, "The Basic Approach": the framework itself and its three founding algorithms.

Setting

Fix a finite index set $I$ of primal variables with non-negative cost coefficients $c_i$ , and a finite index set $J$ of covering constraints, revealed one at a time in the order enumerated by $J$ . Each constraint $j$ is given by a set $S(j) \subseteq I$ (the book's simplified setting, in which every non-zero coefficient equals $1$ and every right-hand side equals $1$ ; Chapter 14 removes this restriction) and asserts $\sum_{i \in S(j)} x_i \ge 1$ . An online covering algorithm may only increase the $x_i$ , never decrease them, and upon a constraint's arrival must eventually make it hold. The online covering problem is to minimize $\sum_i c_i x_i$ subject to every revealed constraint, online. Its Lagrangian dual is the online packing problem: a dual variable $y_j$ arrives together with constraint $j$ , may only be increased while $j$ is being processed, and the objective is to maximize $\sum_j y_j$ subject to $\sum_{j \mid i \in S(j)} y_j \le c_i$ for every $i$ — the packing constraint on $i$ becomes fully known only once every $j$ with $i \in S(j)$ has arrived, so it, too, is revealed gradually. $d := \max_j |S(j)|$ , the largest constraint size, is carried as an explicit parameter throughout.

Section 4.2 gives three algorithms solving both problems simultaneously — the same run produces a covering solution $x$ and a packing solution $y$ — with the same worst-case guarantee but different flavors: Algorithm 1 is a discrete process (each processing round performs a whole number of identical multiplicative-plus-additive updates until its constraint is satisfied); Algorithm 2 is the continuous limit of Algorithm 1 (dual variables increase continuously and $x_i$ follows an explicit exponential of the accumulated dual sum); Algorithm 3 replaces the continuous update with one triggered by an approximate complementary-slackness condition, at the cost of a mild dual infeasibility. All three make essential use of the online order: an algorithm that saw the whole instance up front would trivially solve the offline LP.

Formalization targets

Theorem 4.3 (Algorithm 3 — the goal, p. 124):

\Big(\forall i,\ \textstyle\sum_{j \mid i \in S(j)} y_j \le c_i(1+\ln d)\Big) \ \wedge\ \Big(\forall x''\text{ feasible},\ \textstyle\sum_i c_i x_i \le 2(1+\ln d)\sum_i c_i x''_i\Big) \ \wedge\ \Big(\forall y''\text{ feasible},\ \textstyle\sum_j y''_j \le 2\sum_j y_j\Big).

Theorem 4.1 (Algorithm 1) and Theorem 4.2 (Algorithm 2, p. 118 and p. 121) are the same three-part guarantee for the other two algorithms, with $\log_2(3d+1)$ in place of $1+\ln d$ for Algorithm 1 (and its packing solution genuinely integral), and with $2\ln(1+d)$ in place of both $2(1+\ln d)$ and $2$ for Algorithm 2 (whose packing solution is exactly feasible, not merely approximately so). The competitive ratios are stated against an arbitrary offline-feasible comparison solution on each side (covering and packing) rather than against an unconstructed LP optimum — the standard weak-duality reformulation of " $c$ -competitive", and the one the book's own proofs (which invoke weak duality directly, never LP optimality) actually establish.

Significance

The framework converts three qualitatively different design intuitions — the ski-rental-style discrete doubling, the continuous primal-dual differential equation, and complementary slackness — into three algorithms with an identical asymptotic guarantee, $\Theta(\log d)$ , matching the $\Omega(\log d)$ (packing) and $\Omega(\log n)$ (covering) lower bounds the book proves in Section 4.3 (Lemmas 4.5-4.6, not part of this mission). This is the load-bearing substrate for the rest of the survey: Chapter 5's online set-cover algorithm, Chapter 13's bounded-allocation problem, and Chapter 14's general packing-covering constraints all restate this chapter's framework locally rather than re-deriving it, and are formalized as separate missions in this series. Formalizing it here, once, with the exact constants each proof establishes, is what lets those missions cite a single faithful statement instead of three independently-drifting restatements. No formal development of this framework was found on the platform as of 2026-09-20 (searches below); this mission is the first.

Difficulty

The central obstacle is not the algebra of any single algorithm's proof — each is a short, self-contained argument — but stating the guarantee for an online process using only its final output. An algorithm is characterized here by the closed-form relation its update rule establishes between the accumulated dual sum and the primal value (e.g., Algorithm 3's $x_i = \min(1, d^{-1}\exp(d_i/c_i - 1))$ once activated, $0$ before), together with primal feasibility as the hypothesis that a run has completed; a formalization that instead handed the algorithm the whole instance in advance, or dropped primal feasibility as a hypothesis, would either trivialize the online promise or make the stated bound simply false. A second difficulty specific to this formalization: Theorem 4.3's own proof bounds the primal cost by splitting it into a piece controlled by the final-state complementary-slackness conditions (immediate from the closed form and the $d$ -bound) and a piece controlled by a derivative/telescoping argument over the continuous accumulation process itself — the latter is a genuinely dynamic fact about the trajectory, not just its endpoint, and is recorded as a documented simplification below rather than folded into the hypotheses, since the goal is a faithful statement, not a proof.

Formalization scope

CoveringInstance I J bundles S : J → Finset I, c : I → ℝ, and d : ℝ with 0 < d and ∀ j, (S j).card ≤ d as explicit hypotheses (never derived as d := ⨆ j, (S j).card, matching the book's own presentation and avoiding a vacuous formalization in which d is chosen after the fact to make the bound trivial). dualSum inst y i := ∑_{j \mid i \in S(j)} y_j. Each algorithm's output is a noncomputable def from the accumulated dual data to ℝ (alg1X, alg2X, alg3X), so that "the algorithm's output" is genuinely a function of its dual trajectory rather than an independently-constrained free variable — ruling out the trivializing formalization in which $x$ and $y$ are unrelated variables merely required to satisfy the conclusion's own inequalities. Reals throughout (ℝ, not ℝ≥0 or ENNReal); Finset.filter realizes " $j$ such that $i \in S(j)$ "; Real.logb 2 and Real.log (natural log) match the book's own log₂ and ln. Reusable beyond this mission: CoveringInstance, dualSum, and the weak-duality-style "competitive against any feasible comparison solution" pattern, which 05-online-set-cover, 13-bounded-allocation, and 14-general-packing-covering are expected to restate locally (per this series' own rule against cross-draft imports between concurrent missions) rather than import directly. Welcome contributions: completing any of the three sorrys, and formalizing the Section 4.3 lower bounds (Lemmas 4.5-4.6) as a follow-on mission.

Selected references

N. Buchbinder, J. Naor. The Design of Competitive Online Algorithms via a Primal-Dual Approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. https://doi.org/10.1561/0400000024
N. Buchbinder. Designing Competitive Online Algorithms via a Primal-Dual Approach. PhD thesis, Tel Aviv University, 2008. https://www.tau.ac.il/~nivb/download/phd-thsis.pdf

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning XIII: The Johnson-Lindenstrauss LemmaTextbook

Motivation

High-dimensional data is often computationally expensive to work with and hard to visualize. The Johnson-Lindenstrauss lemma answers a striking question in this setting: can any finite set of points, no matter how high-dimensional the ambient space, be squeezed into a space of dimension depending only on the number of points (logarithmically) and the desired accuracy, while barely disturbing the distances between them? The answer is yes, and — remarkably — a single, data-independent random construction (a random Gaussian matrix) achieves it with positive probability for any point set. This mission formalizes the lemma and the two probabilistic results its proof is built from.

Setting

For a set $V$ of $m$ points in $\mathbb R^N$ , a map $f:\mathbb R^N\to\mathbb R^k$ is a $(1\pm\epsilon)$ -distance-preserving embedding of $V$ if $(1-\epsilon)\|u-v\|^2\le\|f(u)-f(v)\|^2\le(1+\epsilon)\|u-v\|^2$ for every $u,v\in V$ . The book's proof constructs $f=A/\sqrt k$ from a random matrix $A\in\mathbb R^{k\times N}$ with i.i.d. standard normal ( $N(0,1)$ ) entries, and argues by the probabilistic method: for a fixed pair of points, Lemma 15.3 shows $f$ preserves their squared distance up to $(1\pm\epsilon)$ with probability at least $1-2e^{-(\epsilon^2-\epsilon^3)k/4}$ , because the ratio $\|f(x)\|^2/\|x\|^2$ (for $x$ the difference of the two points) is exactly a $\chi^2_k$ random variable, whose two-sided concentration around its mean $k$ is Lemma 15.2. A union bound over the $O(m^2)$ pairs in $V$ then shows the probability that every pair is simultaneously preserved is still strictly positive — so a map with the desired property must exist, even though no single fixed matrix is exhibited.

Formalization targets

Lemma 15.2 (Chi-squared concentration, milestone). If $Q\sim\chi^2_k$ , then for $0<\epsilon<1/2$ , $P[(1-\epsilon)k\le Q\le(1+\epsilon)k]\ge1-2e^{-(\epsilon^2-\epsilon^3)k/4}$ .

Lemma 15.3 (Gaussian random projection distortion, milestone). For $x\in\mathbb R^N$ , $k<N$ , and $A\in\mathbb R^{k\times N}$ with i.i.d. $N(0,1)$ entries,

P\Big[(1-\epsilon)\|x\|^2\le\big\|\tfrac1{\sqrt k}Ax\big\|^2\le(1+\epsilon)\|x\|^2\Big] \ge1-2e^{-(\epsilon^2-\epsilon^3)k/4}.

Lemma 15.4 — the mission's goal (Johnson-Lindenstrauss). For $0<\epsilon<1/2$ , integer $m>4$ , and $k=20\log(m)/\epsilon^2$ , any set $V$ of $m$ points in $\mathbb R^N$ admits a map $f:\mathbb R^N\to\mathbb R^k$ with $(1-\epsilon)\|u-v\|^2\le\|f(u)-f(v)\|^2\le(1+\epsilon) \|u-v\|^2$ for all $u,v\in V$ .

Significance

This is one of the cleanest instances in the book of the probabilistic method: existence is proved without exhibiting the object, by showing a random construction succeeds with positive probability. The target dimension $k=O(\log m/\epsilon^2)$ is independent of the ambient dimension $N$ — the embedding works no matter how large the original feature space is, which is exactly why the lemma has inspired random-projection methods throughout dimensionality reduction, streaming algorithms, and compressed sensing. Prior art on the Prove2Me platform is not faithful here: HighDimProb.Isoperimetry.johnson_lindenstrauss (Vershynin series) proves a Johnson-Lindenstrauss-type bound, but via a uniformly-random $m$ -dimensional subspace and its orthogonal projection, rescaled by $\sqrt{n/m}$ — a genuinely different construction from this book's explicit i.i.d.-Gaussian-matrix map $f=A/\sqrt k$ , and with different (existentially quantified, unspecified) constants rather than this book's explicit $k=20\log(m)/\epsilon^2$ . The two are classically known to give essentially the same qualitative guarantee, but showing them equivalent would require an independent equivalence proof neither platform item nor this mission undertakes; per the chunk's own brief, the Vershynin theorem is cited here only as related work, not reused as a kind: reference item.

Not formalized here: Theorem 15.1 (the PCA solution), the chapter's other headline result. BRIEF.md explicitly flags PCA and JL as the chapter's two independent capstones, not a proof chain, and recommends dropping PCA if the mission focuses purely on JL. Theorem 15.1's proof is an Eckart-Young-type Frobenius-norm optimization argument over the set of rank- $k$ orthogonal projection matrices — sharing no definitions, hypotheses, or proof technique with the probabilistic-method argument this mission's three items are built from. Formalizing it would mean standing up a second, unrelated piece of mathematics (constrained matrix optimization, singular value decomposition) from scratch for a single additional item; per the captain brief's guidance to leave out, rather than approximate, material disproportionate to the time budget, it is omitted. §15.2 (kernel PCA) and §15.3 (Isomap, LLE, Laplacian eigenmaps) are likewise out of scope, per BRIEF.md's own page-range restriction — applications- heavy manifold-learning material with no numbered result feeding Lemma 15.4's proof.

Difficulty

Lemma 15.2's proof is a genuine Chernoff-bound argument: apply Markov's inequality to $\exp(\lambda Q)$ , substitute the chi-squared distribution's own moment-generating function $(1-2\lambda)^{-k/2}$ , and optimize the resulting bound over $\lambda\in(0,1/2)$ by calculus (the book's own choice $\lambda=\epsilon/(2(1+\epsilon))$ is exactly the minimizer) — not a one-line union of tail bounds, and the same argument must be repeated (with a sign flip) for the lower tail before combining via a union bound. Lemma 15.3's proof needs the non-obvious observation that $T_j=\widehat x_j/\|x\|$ (for $\widehat x=Ax$ ) are themselves i.i.d. standard normal — a consequence of $A$ 's i.i.d. Gaussian entries and $\mathbb E[\widehat x_j^2]=\|x\|^2$ , not immediate from the definitions alone — before the sum of their squares can be recognized as exactly a $\chi^2_k$ variable and Lemma 15.2 applied. Lemma 15.4's own proof, while conceptually simple (a single union bound), needs the specific numerical accounting $2m^2e^{-(\epsilon^2-\epsilon^3)k/4}=2m^{5\epsilon-3}<2m^{-1/2}$ at the chosen $k=20\log(m)/\epsilon^2$ and $\epsilon\le1/2$ to conclude the success probability is strictly positive — a formalization that merely asserted "the union bound gives a positive probability" without this exact accounting would not establish the book's specific, tight constant $k=20\log(m)/\epsilon^2$ .

Formalization scope

SqNorm is the coordinate-wise squared Euclidean norm (∑ᵢ vᵢ²), used in place of Mathlib's EuclideanSpace/PiLp norm typeclass machinery to keep the statements close to the book's own concrete ℝ^N/ℝ^k-coordinate notation. IsChiSquaredMGF states the moment-generating-function characterization of a chi-squared distribution that the proof of Lemma 15.2 itself cites (Eq. C.25), rather than restating the book's own Definition C.7, which sits in an out-of-chapter appendix not read for this mission; this is checked (in SELF_REVIEW.md) to be a faithful, uniquely-determining substitute, since a real random variable's law is determined by its MGF wherever it is finite near 0. IsIIDStandardGaussianMatrix states "i.i.d. standard normal entries" via Mathlib's gaussianReal 0 1 (marginal law) together with iIndepFun (joint independence). The goal theorem's target dimension k is taken as ⌈20 log(m)/ε²⌉ (Nat.ceil) rather than the book's real-valued 20 log(m)/ε², since a map's codomain dimension must be a natural number; this rounding only strengthens (never weakens) the existential conclusion, since the union-bound success probability is monotonically increasing in k. No numerical constant in any of the three items is otherwise altered from the book's own displayed form. A trivializing formalization this mission avoids: stating Lemma 15.4 with an unspecified k = O(log m/ε²) (an asymptotic, not an explicit constant) — per BRIEF.md's own naming of this as one of the series' "explicit-constant" results, k's exact formula 20 log(m)/ε² (up to the ceiling needed for well-typedness) is stated directly.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 15, §15.4.
W. B. Johnson, J. Lindenstrauss, "Extensions of Lipschitz mappings into a Hilbert space," Contemporary Mathematics 26, 1984, 189-206.
S. Vempala, The Random Projection Method, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 2004.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning IX: Ranking and the Margin BoundTextbook

Motivation

Ranking is the learning problem behind search engines, recommendation systems and fraud-alert triage: what matters is not a single classification decision but the relative order the system assigns to a set of items, because a user or analyst can only act on the very top of a ranked list. Chapter 10 develops margin-based generalization theory for the score-based ranking setting, transplanting chunk 05-svm's single-sample Rademacher-complexity machinery to a genuinely two-sample structure: a ranking example is a pair of points, one drawn from each of two positions, and the chapter's bound must therefore control two marginal complexities rather than one. It also introduces RankBoost, the ranking analogue of AdaBoost, with a boosting-style empirical-error guarantee proved by the same normalization-factor telescoping argument as chunk 07's AdaBoost bound, adapted to RankBoost's own pairwise per-round quantities.

Setting

A ranking example is a pair (x,x') drawn from a distribution D over X×X, labeled by a preference function f; restricted to {-1,+1} labels (the simplification §10.2 adopts), a scoring function h:X→ℝ misranks (x,x') when f(x,x')(h(x')-h(x)) ≤ 0 (Eq. 10.1/10.2). The empirical margin loss R̂_{S,ρ}(h) (Eq. 10.3) uses the same Φ_ρ (Definition 5.5) as chunk 05-svm, restated locally here. Writing S1, S2 for the two coordinate projections of a pair sample S, and D1, D2 for the corresponding marginals of D, R_m^{D1}(H) and R_m^{D2}(H) are the Rademacher complexities of H under each marginal (p. 241). Theorem 10.1 bounds R(h) in terms of these two Rademacher-complexity terms, both in their population form (R_m^{D1}, R_m^{D2}) and their empirical form (R̂_{S1}, R̂_{S2}), via chunk 03's Theorem 3.3 applied through an auxiliary hypothesis family H̃ = {((x,x'),y) ↦ y[h(x')-h(x)]}. Corollary 10.2 specializes this to kernel-based linear scoring hypotheses; §10.4 introduces RankBoost (Figure 10.1), whose per-round weighted pairwise-outcome fractions ε_t^+, ε_t^- (Eq. 10.11) play the role AdaBoost's single ε_t plays in chunk 07, and Theorem 10.3 bounds RankBoost's empirical error in terms of them. Corollary 10.4 combines Theorem 10.1 with Lemma 7.4 (the convex hull of H has the same empirical Rademacher complexity as H, restated as a standing fact of the boosting series) to give RankBoost's own margin-based guarantee.

Formalization targets

Theorem 10.1 — the mission's goal. For H a set of real-valued functions, ρ>0, δ>0, with probability at least 1-δ, for all h∈H:

R(h) \le \hat R_{S,\rho}(h) + \tfrac2\rho(R_m^{D_1}(H)+R_m^{D_2}(H)) + \sqrt{\tfrac{\log(1/\delta)}{2m}}

R(h) \le \hat R_{S,\rho}(h) + \tfrac2\rho(\hat R_{S_1}(H)+\hat R_{S_2}(H)) + 3\sqrt{\tfrac{\log(2/\delta)}{2m}}.

Corollary 10.2 (milestone). For a PDS kernel K with r an upper bound on K(x,x), feature map Φ, and H = {x↦w·Φ(x) : ‖w‖≤Λ}, fixed ρ>0: R(h) ≤ R̂_{S,ρ}(h) + 4√(r²Λ²/ρ²/m) + √(log(1/δ)/(2m)).

Theorem 10.3 (milestone). RankBoost's empirical error verifies R̂_S(f) ≤ exp(-2∑_t((ε_t^+-ε_t^-)/2)²), and ≤ exp(-2γ²T) if the edge is uniformly at least γ>0.

Corollary 10.4 (milestone). Theorem 10.1's first bound, applied to h∈conv(H).

Significance

Theorem 10.1's proof is the chapter's genuine new technique, not a restatement of chunk 05's Theorem 5.8: the two-sample decomposition (splitting the supremum over H̃ into a term on x' alone and a term on x alone, each bounded by the Rademacher complexity under its own marginal) is what the 2/ρ · (R_m^{D1}+R_m^{D2}) structure expresses, and collapsing it to a single-sample bound would either be false or silently assume D1=D2 (which only holds for a symmetric D, an assumption the theorem does not make). Corollary 10.2 is the direct theoretical basis for the ranking SVM algorithm §10.3 derives. Theorem 10.3 mirrors chunk 07-boosting's Theorem 7.2 almost line for line in its proof technique (the same telescoping product of normalization factors Z_t), but with genuinely different per-round quantities (ε_t^+, ε_t^- rather than a single ε_t) that must not be conflated with AdaBoost's own, per BRIEF.md's pitfall note. Corollary 10.4 is what makes RankBoost's output (a linear, not convex, combination — normalized by ‖α‖_1) provably generalize independently of the number of boosting rounds T, the ranking analogue of chunk 07's Corollary 7.5. No prior art exists on the platform: GET /theorems?q=ranking%20loss returns zero hits.

Difficulty

Theorem 10.1's proof needs the two-sample structure carried through explicitly: H̃'s Rademacher complexity splits, via the sub-additivity of sup and the fact that y_iσ_i and σ_i have the same distribution, into a term on S2 alone and a term on S1 alone — treating a ranking sample as an ordinary single sample (chunk 03's single-hypothesis-set machinery applied naively) would drop this structure entirely and is exactly the pitfall BRIEF.md names. Theorem 10.3's proof requires Z_t = ε_t^0 + 2√(ε_t^+ε_t^-) be bounded via the identity 4ε_t^+ε_t^- = (1-ε_t^0)^2 - (ε_t^+-ε_t^-)^2 and the inequality 1-x ≤ e^{-x} — the same telescoping-normalizer technique as AdaBoost's Theorem 7.2, but RankBoost's own D_t, ε_t^+, ε_t^- genuinely differ (they are defined via pairwise outcomes y_i(h(x'_i)-h(x_i)) ∈ {-1,0,+1}, not a single-point disagreement h(x_i)≠y_i) and must be modeled as their own recursively-defined algorithm state, not obtained by substitution into chunk 07's AdaBoost Lean.

Formalization scope

MarginLossFunction restates chunk 05-svm's Definition 5.5; EmpiricalRademacherComplexity/ RademacherComplexity restate chunk 03-rademacher-vc's Definitions 3.1/3.2; IsPDS restates chunk 06-kernels's PDS-kernel definition; ConvHull restates chunk 07-boosting's convex-hull definition — all duplicated rather than imported since a draft item cannot import another chunk's draft module, and none is listed as reusable in missions/README.md's "Published definitions" table at the time of this session. D1, D2 are computed directly as Measure.map Prod.fst D/Measure.map Prod.snd D rather than posited via a separate marginal hypothesis, so the theorem statement itself pins down that they are genuinely the marginals of the sampling distribution D, not independent parameters. Corollary 10.2's r is an explicit upper bound on K(x,x) (hrK : ∀ x, K x x ≤ r) rather than a literal sSup, per BRIEF.md's pitfall note about the possibly-infinite supremum — the theorem's conclusion is monotonic in r, so this is not a weakening. RankBoost's D_t, ε_t^+, ε_t^-, α_t, Z_t and returned function f are modeled as their own recursively-defined algorithm state (mirroring chunk 07's AdaBoostDist/AdaBoostEpsilon/AdaBoostEnsemble pattern exactly, but built from RankBoost's own pairwise-outcome quantities, never by substituting into the AdaBoost Lean, per BRIEF.md's pitfall note) — RankBoostEpsilonPlus/RankBoostEpsilonMinus are already tied to RankBoost's own D_t and selected base ranker, so Theorem 10.3 needs no separate hypothesis connecting them (the same trivialization guard chunk 07's own Theorem 7.2 documents). No numerical constant is altered from the book in any of the four theorems.

Not formalized: §10.3's ranking-SVM primal/dual optimization problems (an algorithm derived from Corollary 10.2, not a generalization-theoretic result); §10.4.2 (RankBoost as coordinate descent, an algorithmic-equivalence argument, not a generalization bound); §10.5 (bipartite ranking, its own distinct problem formulation with a different generalization error, Eq. 10.20, explicitly out of scope per BRIEF.md); §10.6-10.7 (preference-based ranking, other criteria), out of scope per BRIEF.md. The uniform-over-ρ extension mentioned after both Theorem 10.1's and Corollary 10.2's proofs (referencing Theorem 5.9's technique from a different chapter) is not drafted, matching chunk 09-multiclass's identical scope decision for the analogous remark.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 10 (§10.1-10.4).
Y. Freund, R. Iyer, R. E. Schapire, Y. Singer, "An efficient boosting algorithm for combining preferences," JMLR 4, 2003 (RankBoost's origin).
C. Cortes, M. Mohri, "AUC optimization vs. error rate minimization," NeurIPS 2003 (the ranking-SVM connection §10.3 develops).

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning IV: Support Vector Machines and the Margin BoundTextbook

Motivation

Support vector machines were, for two decades, the workhorse of applied classification, and the reason offered for their success was always geometric: SVMs maximize the margin between the two classes. Chapter 3's VC-dimension bound cannot explain why this should help — for linear hypotheses in $\mathbb R^N$ its bound depends on $N+1$ and is uninformative whenever the feature dimension is large relative to the sample size, exactly the regime (kernel-induced or high-dimensional features) where SVMs are most often used. Chapter 5 answers the question this leaves open: a generalization bound for a real-valued hypothesis, stated in terms of its margin on the training sample, that does not depend on the ambient dimension at all. This is also the template every later chapter's margin bound specializes (multi-class classification, ranking, and, indirectly, boosting all reuse the same Rademacher-complexity-of-a-Lipschitz-loss argument developed here).

Setting

A hypothesis here is a real-valued function $h:X\to\mathbb R$ , not (as in Chapters 2-3) a function into $\{-1,+1\}$ : for a labeled point $(x,y)$ with $y\in\{-1,+1\}$ , the sign of $h(x)$ gives the prediction and $|h(x)|$ is read as the classifier's confidence. The confidence margin of $h$ at $(x,y)$ is $y\,h(x)$ ; it is positive exactly when $h$ classifies $x$ correctly. For $\rho>0$ , the $\rho$ -margin loss $\Phi_\rho:\mathbb R\to\mathbb R$ (Definition 5.5) is

\Phi_\rho(x) = \min\Big(1,\max\Big(0,1-\frac x\rho\Big)\Big),

equal to $1$ when $x\le 0$ (misclassified), $0$ when $x\ge\rho$ (classified with confidence at least $\rho$ ), and interpolating linearly in between; it is $1/\rho$ -Lipschitz. The empirical margin loss on a sample $S=(x_1,\dots,x_m)$ with labels $y_1,\dots,y_m$ (Definition 5.6) is $\hat R_{S,\rho}(h) = \frac1m\sum_{i=1}^m\Phi_\rho(y_ih(x_i))$ — the fraction of training points misclassified or classified with confidence below $\rho$ , a strictly stronger requirement than plain misclassification. The (population) generalization error is $R(h)=\Pr_{(x,y)\sim D}[y\,h(x)\le 0]$ . Rademacher complexity, $\hat R_S(H)$ and $R_m(H)$ (Definitions 3.1-3.2, restated here since chunk 03-rademacher-vc's own copies are still drafts), measure how well a real-valued hypothesis class $H$ correlates with random sign noise on a sample, and are the vehicle through which the margin bound's complexity term is expressed.

Formalization targets

Lemma 5.7 (Talagrand's lemma, milestone). For $l$ -Lipschitz $\Phi_1,\dots,\Phi_m:\mathbb R\to\mathbb R$ and any hypothesis set $H$ of real-valued functions,

\frac1m\,\mathbb E_\sigma\Big[\sup_{h\in H}\sum_{i=1}^m\sigma_i(\Phi_i\circ h)(x_i)\Big] \le l\,\hat R_S(H).

Theorem 5.10 (Rademacher complexity of bounded-norm linear hypotheses, milestone). For $S\subseteq\{x:\|x\|\le r\}$ and $H=\{x\mapsto w\cdot x:\|w\|\le\Lambda\}$ ,

\hat R_S(H) \le \sqrt{r^2\Lambda^2/m}.

Corollary 5.11 (margin bound for linear hypotheses, milestone). For the same $H$ and $X\subseteq\{x:\|x\|\le r\}$ , fixing $\rho>0$ , with probability at least $1-\delta$ ,

R(h) \le \hat R_{S,\rho}(h) + 2\sqrt{\frac{r^2\Lambda^2/\rho^2}{m}} + \sqrt{\frac{\log(1/\delta)}{2m}} \quad\text{for all } h\in H.

Theorem 5.8 — the mission's goal. For any set $H$ of real-valued functions and $\rho>0$ , with probability at least $1-\delta$ , both

R(h) \le \hat R_{S,\rho}(h) + \frac2\rho R_m(H) + \sqrt{\frac{\log(1/\delta)}{2m}}

R(h) \le \hat R_{S,\rho}(h) + \frac2\rho \hat R_S(H) + 3\sqrt{\frac{\log(2/\delta)}{2m}}

hold simultaneously for all $h\in H$ .

Significance

Theorem 5.8 is genuinely dimension-free: unlike Chapter 3's VC-dimension bound (5.36 in the book, restated from Corollary 3.19), it holds regardless of the ambient feature dimension $N$ , depending instead only on the hypothesis class's Rademacher complexity and the chosen margin $\rho$ . Specialized to bounded-norm linear hypotheses (Corollary 5.11), this gives the theoretical justification most often cited for SVMs and every other margin-maximization algorithm: whenever the training data admits a large geometric margin, the empirical margin loss at that margin is small (often zero, in the separable case) and the bound is tight regardless of $N$ . Every later chapter's own margin bound (multi-class in Chapter 9, ranking in Chapter 10) is a direct structural descendant of Theorem 5.8's proof technique. No prior art on the Prove2Me platform is faithful: GET /theorems?q=support+vector+machine and q=margin+bound return no hits on this book's model (the two q=margin+bound hits found, both from the Aether Catalog, state a different comparison — a VC-type bound is eventually worse than a fixed Rademacher-type bound as a function of dimension — not Theorem 5.8 itself); q=Talagrand and q=contraction+principle return several hits (Ledoux/Talagrand convex-distance concentration, Rudin's Banach-space contraction-mapping theorem, a generic Rademacher-sign contraction lemma for quadratic sums) but every one states either a different mathematical object (metric-space fixed points, Talagrand's concentration inequality on product spaces) or a different idiom (squared vs. linear coordinate sums) from Lemma 5.7's function-composition contraction — none reused. All nine items are drafted fresh.

Not formalized here: Theorem 5.4 (the SVM sparsity/leave-one-out bound). It is listed as a candidate milestone in BRIEF.md, but its statement and proof depend on the primal/dual SVM optimization problem itself (the Lagrangian, KKT conditions, and the resulting definition of a "support vector" as a training point with nonzero dual coefficient) — a materially different, non-margin-based proof technique (leave-one-out stability of the trained hypothesis, via Lemma 5.3) that shares no definitions with the margin-bound family this mission's goal and other milestones are built on. Formalizing it faithfully would require standing up the SVM primal/dual formalism (Lagrangian, complementary slackness, the "support vector" predicate itself) from scratch, which is disproportionate to a single additional milestone within this mission's budget; per the captain brief's guidance to leave out, rather than approximate, a statement that cannot be made faithful in the time available, it is omitted.

Difficulty

The proof of Theorem 5.8 needs the empirical margin loss's zero-one-loss upper bound ( $\mathbb 1_{u\le 0}\le\Phi_\rho(u)$ ) applied before invoking Theorem 3.3's Rademacher generalization bound on the composed class $\tilde{\tilde H}=\{\Phi_\rho\circ f: f\in\tilde H\}$ , $\tilde H=\{(x,y)\mapsto y\,h(x):h\in H\}$ — reversing this order (bounding $R(h)$ by a Rademacher complexity computed on the zero-one loss directly) does not work, because the zero-one loss is not Lipschitz. Talagrand's lemma is exactly what lets the $1/\rho$ -Lipschitz surrogate $\Phi_\rho$ be pulled outside the Rademacher complexity, at the cost of a factor $1/\rho$ and no worse; its own proof is an induction removing one Rademacher variable at a time, using a two-point supremum argument (fixing $\epsilon>0$ , choosing near-optimal $h_1,h_2$ ) that does not simplify to anything less than genuine care with suprema of non-smooth objects — a formalization attempting to replace this with a naive linearity-of-expectation argument would be proving a false or vacuous statement, since $\sup$ does not commute with linear combinations. Theorem 5.10's bound needs the Cauchy-Schwarz and Jensen inequalities used in the particular order the book uses them (Cauchy-Schwarz on the empirical sup, then Jensen on the expectation of a norm, then the independence of the $\sigma_i$ s) — the bound $\hat R_S(H)\le r\Lambda/\sqrt m$ does not follow from either inequality alone.

Formalization scope

EmpiricalRademacherComplexity/RademacherComplexity are restated locally in SVM, byte-identical to chunk 03-rademacher-vc's own copies (a draft item cannot import another chunk's draft module); this duplication collapses once 03-rademacher-vc is uploaded and listed in missions/README.md's "Published definitions" table. MarginGeneralizationError is a new, real-valued-hypothesis specialization of Definition 2.1 (R(h) = P[y h(x) ≤ 0]), distinct from every earlier chunk's {-1,+1}-valued GeneralizationError, since no earlier chunk's own copy matches this chapter's real-valued convention. PhiRho/EmpiricalMarginLoss are new. The goal theorem (margin_bound_binary_classification) states H's own Rademacher complexity computed on the marginal X-distribution (D.map Prod.fst), matching the book's final displayed form — the proof's intermediate step (the lifted class $\tilde H=\{(x,y)\mapsto y h(x)\}$ having the same Rademacher complexity as $H$ itself, since $y\in\{-1,+1\}$ ) is not separately drafted, only the theorem's statement. Theorem 5.10/Corollary 5.11 generalize the book's ambient $\mathbb R^N$ to an arbitrary real inner-product space X ([NormedAddCommGroup X] [InnerProductSpace ℝ X]), a harmless generalization since the book's proof (Cauchy-Schwarz, Jensen, orthogonality of Rademacher signs) uses only the inner-product structure, never finite dimension; [BorelSpace X] is added to Corollary 5.11's statement to make the measurable structure under which MarginGeneralizationError is well-posed explicit, since every x ↦ ⟨w, x⟩ is automatically Borel-measurable — not a substantive restriction, the book never discusses measurability of linear functionals. No numerical constant in any of the four theorems is altered from the book's own displayed form. A trivializing formalization this mission avoids: stating Theorem 5.8 only for a Finset/finite H (which would make it a disguised instance of Chapter 2's finite-hypothesis bound rather than the chapter's genuinely new, complexity-based argument) — H : Set (X → ℝ) is left fully general, exactly as the book states it.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 5.
C. Cortes, V. Vapnik, "Support-vector networks," Machine Learning 20(3), 1995, 273-297.
M. Talagrand, "Sharper bounds for Gaussian and empirical processes," The Annals of Probability 22(1), 1994, 28-76.
P. Bartlett, S. Mendelson, "Rademacher and Gaussian complexities: risk bounds and structural results," Journal of Machine Learning Research 3, 2002, 463-482.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning II: Rademacher Complexity and VC-DimensionTextbook

Motivation

Chapter 2's finite-hypothesis-set learning bound is uninformative the moment $H$ is infinite — $\log|H|$ diverges — yet most hypothesis sets used in practice (linear separators, neural networks, decision trees) are infinite. Chapter 3 answers the question the previous chapter's own worked example (axis-aligned rectangles, Example 2.4) leaves open: is efficient learning from a finite sample still possible for an infinite hypothesis set, and can this be shown in general rather than case by case? The chapter's answer runs through two complementary notions of complexity — Rademacher complexity, a data-dependent measure of how well a function family correlates with random noise, and the VC-dimension, a purely combinatorial measure of the number of distinct labelings a hypothesis set can realize on a finite point set — connected by Massart's lemma and Sauer's lemma, and culminating in a generalization bound that replaces $\log|H|$ with the VC-dimension $d$ .

Setting

For a family $G$ of functions $Z\to[0,1]$ and a sample $S=(z_1,\dots,z_m)$ , the empirical Rademacher complexity $\hat R_S(G) = \mathbb E_\sigma[\sup_{g\in G}\frac1m\sum_i\sigma_i g(z_i)]$ (Definition 3.1) measures how well $G$ fits random sign noise $\sigma$ on $S$ ; the Rademacher complexity $R_m(G) = \mathbb E_{S\sim D^m}[\hat R_S(G)]$ (Definition 3.2) averages this over samples. Theorem 3.3 converts a Rademacher-complexity bound directly into a generalization bound via McDiarmid's inequality. For binary hypothesis sets $H\subseteq(X\to\{-1,+1\})$ , the growth function $\Pi_H(m)$ (Definition 3.6) counts the maximum number of distinct dichotomies $H$ realizes on $m$ points, and the VC-dimension $\mathrm{VCdim}(H)$ (Definition 3.10) is the largest $m$ for which $\Pi_H(m)=2^m$ (i.e. $H$ shatters some set of $m$ points). Massart's lemma (Theorem 3.7) is the purely combinatorial tool bounding the expected maximum of a sum of signed vector components by $\sqrt{\log|A|}$ , and Sauer's lemma (Theorem 3.17) bounds the growth function itself, by induction on $m+d$ , whenever the VC-dimension is finite.

Formalization targets

Theorem 3.3 (Rademacher generalization bound, milestone). For $G:Z\to[0,1]$ and any $\delta>0$ , with probability at least $1-\delta$ over an i.i.d. sample $S$ of size $m$ , for all $g\in G$ : $\mathbb E[g(z)] \le \frac1m\sum_i g(z_i) + 2R_m(G) + \sqrt{\log(1/\delta)/(2m)}$ .

Theorem 3.7 (Massart's lemma, milestone). For a finite $A\subseteq\mathbb R^m$ with $r=\max_{x\in A}\|x\|_2$ : $\mathbb E_\sigma[\frac1m\sup_{x\in A}\sum_i\sigma_i x_i] \le r\sqrt{2\log|A|/m}$ .

Theorem 3.17 (Sauer's lemma, milestone). For $H$ with $\mathrm{VCdim}(H)=d$ , for all $m\in\mathbb N$ : $\Pi_H(m) \le \sum_{i=0}^d\binom{m}{i}$ .

Corollary 3.19 — the mission's goal. For $H\subseteq(X\to\{-1,+1\})$ with $\mathrm{VCdim}(H)=d$ and any $\delta>0$ , with probability at least $1-\delta$ , for all $h\in H$ :

R(h) \le \hat R_S(h) + \sqrt{\frac{2d\log(em/d)}{m}} + \sqrt{\frac{\log(1/\delta)}{2m}}.

Significance

Corollary 3.19 is the chapter's answer to the question chapter 2 leaves open: it is Theorem 2.13's direct infinite-hypothesis-set generalization, replacing $\log|H|$ (undefined for infinite $H$ ) with the VC-dimension $d$ (finite even for many infinite hypothesis sets, such as halfspaces in $\mathbb R^k$ , which have VC-dimension $k+1$ ). It is also the template every later margin bound in the book specializes (Chapters 5, 9, 10's SVM, multi-class and ranking margin bounds all replace this bound's uniform $\log|H|$ /VC-dimension term with a scale-sensitive complexity measure derived from the same Rademacher-complexity machinery), and Sauer's lemma is independently one of the most cited results in learning theory and extremal combinatorics. No prior art on the Prove2Me platform is faithful to any of this chapter's content: RademacherSymmetrization.radS_chernoff (Aether Catalog) proves a different, Massart-optimized Chernoff bound for the empirical Rademacher complexity of a finite class — a different object (empirical vs. population) with a different bound form from Theorem 3.3/3.5 — and sauerShelah_full proves only the trivial identity sauerShelahBound k k = 2^k, not Sauer's lemma itself. A further hit, sauer_shelah (Aether Catalog, Algebra/SauerShelah.lean), does state the Sauer-Shelah bound itself (F.card ≤ ∑_{i≤d} C(n,i) for a family F of subsets of Fin n shattering no set larger than d) — checked and not reused: it is a different idiom from Theorem 3.17 as this chunk needs it, a fixed finite ambient domain Fin n with F a Finset of its subsets directly, rather than the book's own growth function Π_H(m) (a supremum over point-tuples drawn from an arbitrary, possibly infinite X, Definition 3.6) that this chunk's other items and the goal (Corollary 3.19) are built on; reusing it would require either abandoning GrowthFunction/HasVCDim (needed faithfully by the goal itself) or a nontrivial reduction lemma this mission's budget does not include, so sauer_lemma is drafted fresh against this chunk's own GrowthFunction/HasVCDim. All ten items are drafted fresh.

Difficulty

Sauer's lemma's proof is a genuine two-parameter induction (on $m+d$ ) with a real combinatorial construction: restricting $H$ to a sample $S$ of size $m$ , then splitting the restricted family into $G_1$ (its restriction to the first $m-1$ points) and $G_2$ (the concepts whose membership in $G$ changes with the addition of the $m$ -th point), with $|G_1|+|G_2|=|G|$ and $\mathrm{VCdim}(G_2) \le \mathrm{VCdim}(G)-1$ — a genuinely combinatorial argument, not a statement that unfolds by simp; a weaker restatement using only the trivial bound $\Pi_H(m)\le 2^d$ would be true but is explicitly not what Theorem 3.17 states (BRIEF.md's named trivializing formalization for this chapter). Massart's lemma needs the expectation of a supremum over a finite set of $2^m$ -many sign patterns kept as an honest average, not silently replaced by a looser union bound. Corollary 3.19's own $em/d$ term inside the logarithm needs the side condition $d\le m$ carried through explicitly — Corollary 3.18's own domain restricts to $m\ge d$ , and the bound is false, not merely unproved, without it (at $m<d$ , $em/d$ can be smaller than $1$ , making the logarithm negative).

Formalization scope

GeneralizationError/EmpiricalError are restated locally in this chunk's RademacherVC namespace (byte-identical in content to chunk 02-pac's own copies), since a draft item cannot import another chunk's draft module; this duplication is expected and will collapse once 02-pac is moderated, uploaded and listed as reusable in missions/README.md's "Published definitions" table. EmpiricalRademacherComplexity/Massart's lemma model the Rademacher signs σ as ranging over the finite type Fin m → Bool rather than a measure-theoretic i.i.d. process, so the "expectation over σ" in both is the exact finite uniform average over its 2^m outcomes — faithful and simpler than a MeasureTheory construction, since σ's distribution really is uniform on a finite set of outcomes for every finite m. GrowthFunction takes a tuple of m points (Fin m → X) rather than a size-m subset of X, a harmless generalization (repeated points never increase the dichotomy count) documented in the item's own docstring. HasVCDim is a Prop parametrized by the candidate dimension rather than a total ℕ/ℕ∞-valued function, so it does not cover the book's VCdim(H)=+\infty case (Examples 3.15-3.16); every theorem using it takes HasVCDim H d as an explicit hypothesis, matching the book's own "let H... with VCdim(H)=d." Theorem 3.3 adds an explicit measurability hypothesis on G (hGm) beyond the book's own displayed statement, needed to keep the Bochner integral ∫ z, g z ∂D from silently evaluating to 0 for a non-measurable g — this is the book's own standing assumption (footnote 3, p. 30) made an explicit hypothesis rather than an implicit one. No numerical constant in any of the four theorems is altered from the book's own; Corollary 3.19's side condition d ≤ m is kept explicit, per BRIEF.md's pitfall note.

Not formalized: Lemma 3.4 and Theorem 3.5 (the binary-classification specialization of Theorem 3.3 via the zero-one-loss identity $\hat R_S(G)=\frac12\hat R_{S_X}(H)$ ), Corollary 3.8 and Corollary 3.9 (the intermediate Rademacher-to-growth-function and growth-function generalization bounds), and Corollary 3.18 (the VC-dimension bound on the growth function, $\Pi_H(m)\le(em/d)^d$ for $m\ge d$ ) — five intermediate results in the proof chain Theorem 3.3 → Theorem 3.5 → Corollary 3.8/3.9 → Sauer's lemma → Corollary 3.18 → Corollary 3.19 that are not independently drafted as milestones, per the budget guidance to keep a chunk to a goal plus its most load-bearing 3-8 milestones rather than every numbered result on the page; the three drafted milestones (Theorem 3.3, Massart's lemma, Sauer's lemma) are the chain's three genuinely distinct proof techniques (McDiarmid's inequality, a probabilistic-maximum bound, and a combinatorial induction), and the goal theorem's own statement is Corollary 3.19 exactly as displayed, not a restatement of any intermediate corollary. Radon's theorem (Theorem 3.13, background for the hyperplane VC-dimension example) and the worked VC-dimension examples (intervals, hyperplanes, rectangles, convex polygons, sine functions) are illustrations, not general results, and are not formalized — drafting only the example computations (e.g. VCdim(hyperplanes) = d+1) instead of the general finite-H machinery is exactly the trivializing formalization this mission avoids.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 3.
V. Vapnik, A. Chervonenkis, "On the uniform convergence of relative frequencies of events to their probabilities," Theory of Probability and its Applications 16(2), 1971, 264-280.
N. Sauer, "On the density of families of sets," Journal of Combinatorial Theory, Series A 13(1), 1972, 145-147.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning I: The PAC Learning FrameworkTextbook

Motivation

How many labeled examples does a learning algorithm need to see before its output generalizes well to unseen data? Chapter 2 of Foundations of Machine Learning answers this question for the simplest nontrivial setting — a finite hypothesis set — and in doing so introduces the book's central object, the Probably Approximately Correct (PAC) learning framework: a distribution-free, high-probability guarantee relating a learner's sample size to the accuracy and confidence of its output. Every later chapter's generalization bound (VC-based, Rademacher-based, margin-based) is a variant of the same "probability of a bad event is small" argument this chapter proves in its most elementary form, so getting the chapter's core definitions and its two bracketing theorems (consistent and inconsistent finite- $H$ ) right is the foundation the rest of the book's guarantees build on.

Setting

A learner sees a sample $S = (x_1,\dots,x_m)$ drawn i.i.d. from a fixed but unknown distribution $D$ on an instance space $X$ , labeled by an unknown target concept $c$ drawn from a concept class $C$ ; a hypothesis $h$ from a fixed hypothesis set $H$ is judged by its generalization error $R(h) = \Pr_{x\sim D}[h(x)\ne c(x)]$ (Definition 2.1) against its empirical error $\hat R_S(h) = \frac1m\sum_i \mathbb 1_{h(x_i)\ne c(x_i)}$ (Definition 2.2) on the observed sample. A concept class is PAC-learnable (Definition 2.3) if some algorithm, given a polynomially-bounded number of samples, returns a hypothesis whose generalization error is at most $\epsilon$ with probability at least $1-\delta$ , for every accuracy $\epsilon$ and confidence $\delta$ and every distribution $D$ — the "distribution-free" and "for all target concepts" character of the definition is what makes it a genuine worst-case learning guarantee rather than an average-case one tailored to a particular data-generating process.

Formalization targets

Theorem 2.5 (consistent case, milestone). If $H$ is finite and algorithm $A$ always returns a hypothesis consistent with the target concept on the training sample ( $\hat R_S(h_S)=0$ ), then $\Pr_{S\sim D^m}[R(h_S)\le\epsilon]\ge1-\delta$ whenever $m \ge \frac1\epsilon(\log|H|+\log\frac1\delta)$ .

Corollary 2.11 (single-hypothesis Hoeffding bound, milestone). For a fixed hypothesis $h:X\to\{0,1\}$ and any $\delta>0$ , with probability at least $1-\delta$ , $R(h) \le \hat R_S(h) + \sqrt{\log(2/\delta)/(2m)}$ .

Theorem 2.13 (inconsistent case, goal). For a finite hypothesis set $H$ and any $\delta>0$ , with probability at least $1-\delta$ , simultaneously for every $h\in H$ ,

R(h) \le \hat R_S(h) + \sqrt{\frac{\log|H|+\log(2/\delta)}{2m}}.

Significance

Theorem 2.13 is the chapter's capstone because it removes Theorem 2.5's consistency requirement — the typical case in practice, where no hypothesis in $H$ perfectly fits the training data — while paying only an additive $\log|H|$ price inside the square root, via a union bound over $H$ applied to Corollary 2.11's per-hypothesis concentration bound. It is also the template every later generalization bound in the book refines: Chapter 3 replaces $\log|H|$ with the growth function / VC-dimension to handle infinite hypothesis sets, and Chapter 3's Rademacher-complexity bound is the direct machine-independent generalization of the same argument. No prior art on the Prove2Me platform is faithful to this chapter's PAC-learning content (GET /theorems?q=PAC-learnable returns no hits), so all six items are drafted fresh.

Difficulty

Theorem 2.13's own proof is a short combination of two ideas already present in the chapter (Corollary 2.11's Hoeffding bound plus a union bound over $|H|$ hypotheses), but each ingredient carries its own faithfulness burden. Corollary 2.11 needs the sample $S$ and the target hypothesis $h$ kept in the right relationship — $h$ fixed, $S$ random — for the bound to be Hoeffding's inequality and not a vacuous statement about a random hypothesis. Theorem 2.13 needs the $\forall h\in H$ quantifier placed inside the probability event (a single sample $S$ must work for every $h$ at once), not outside it (which would only assert each $h$ 's bound holds with high probability for a sample chosen depending on $h$ ) — the difference between a uniform convergence bound and $|H|$ separate, weaker statements. Definition 2.3's "polynomial function $\mathrm{poly}(\cdot,\cdot,\cdot,\cdot)$ " is a genuine formalization judgment call, addressed below.

Formalization scope

GeneralizationError/EmpiricalError are typed generally over $X, Y$ (matching Definition 2.1/2.2's own general statement, " $h : X\to Y$ "), since Theorem 2.5 itself is stated for general $Y$ , not just $Y=\mathrm{Bool}$ ; Corollary 2.11 and Theorem 2.13 specialize to $h : X\to\mathrm{Bool}$ , matching their own explicit " $h:X\to\{0,1\}$ " (Corollary 2.11) and the surrounding inconsistent-case section's restriction to binary classification. The i.i.d. sample $S\sim D^m$ is modeled as the identity random variable on the product-measure space $(\mathrm{Fin}\ m \to X,\ \mathrm{Measure.pi}(\lambda\_.\ D))$ in both Corollary 2.11 and Theorem 2.13, matching the book's own $S\sim D^m$ notation exactly. IsPACLearnable (Definition 2.3) makes "polynomial function" precise as a function bounded above by $K\cdot(a+b+n+s+1)^k$ for some constants $K>0$ , $k\in\mathbb N$ , uniform in its (nonnegative) arguments — the standard reading of "polynomial in its arguments" in the absence of a ready-made multivariate polynomial-growth predicate in Mathlib; dropping this constraint entirely (stating only "there is some threshold function") would silently weaken Definition 2.3 to a strictly easier notion of learnability, since virtually any finite or well-behaved concept class admits some (possibly super-polynomial) sample-complexity threshold — this is exactly the distinction Example 2.7 (the universal concept class) uses to demonstrate a class that is not PAC-learnable despite admitting a consistent hypothesis set. No numerical constant in Theorem 2.5, Corollary 2.11 or Theorem 2.13 is altered from the book's own; no upper bound on $\delta$ is added anywhere the book itself leaves it unrestricted (the theorems remain true, if vacuous, for $\delta>1$ ). Not formalized: the "efficiently PAC-learnable" running-time clause of Definition 2.3 (a second, independent polynomial-time condition on $A$ not needed by either milestone or the goal); Corollary 2.10 (the raw two-sided Hoeffding statement Corollary 2.11 is immediately derived from by solving for $\epsilon$ , making it redundant with Corollary 2.11 as a formalization target); the axis-aligned-rectangles worked example (Example 2.4–2.9), which illustrates the framework rather than proving a new general result, and the trivializing formalization this chapter invites — reusing Mathlib's rectangle machinery to encode only the specific two-dimensional geometric argument rather than the general finite- $H$ theorems — is exactly what this mission avoids by drafting Theorems 2.5 and 2.13 in their general, hypothesis-set-agnostic form.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 2.
W. Hoeffding, "Probability inequalities for sums of bounded random variables," Journal of the American Statistical Association 58(301), 1963, 13-30.

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Combinatorics Complexity Theory·Captain: Lucas

4-to-1 Games with Perfect CompletenessResearch Paper

Motivation

Many approximation problems resist the standard PCP toolkit: the best known NP-hardness factors for Max-Cut, Vertex-Cover and approximate graph colouring are far from the best known polynomial-time algorithms. To explain this gap, Khot (CCC 2002) proposed the Unique-Games Conjecture and the family of $d$ -to-1 Games Conjectures. The $d$ -to-1 conjectures assert perfect completeness: the hard instances are either fully satisfiable, or satisfiable only to a vanishing extent. Perfect completeness is what makes these conjectures usable for colouring problems, where a "yes" instance must be genuinely $3$ -colourable rather than almost so.

A line of work culminating in Khot–Minzer–Safra and Dinur–Khot–Kindler–Minzer–Safra established the almost-perfect completeness version for $2$ -to-1 games: for every $\varepsilon>0$ there is an alphabet bound $r$ such that distinguishing value $\ge 1-\varepsilon$ from value $\le\varepsilon$ is NP-hard. Their route goes through Håstad's hardness for linear equations, which cannot have perfect completeness, so the loss is intrinsic to the technique. The source paper of this mission removes that loss for $d = 4$ .

Setting

A label-cover instance $\Psi$ (Definition 1.1 of the source) consists of a bipartite graph $G = (L \sqcup R, E)$ , two finite alphabets $\Sigma_L, \Sigma_R$ , and for each edge $e = (u,v)$ a constraint $\Phi_e \subseteq \Sigma_L \times \Sigma_R$ . The constraint is a projection constraint if there is $\varphi_e : \Sigma_L \to \Sigma_R$ with $\Phi_e = \{(\sigma, \varphi_e(\sigma))\}$ , and a $d$ -to-1 constraint if in addition $|\varphi_e^{-1}(\sigma)| = d$ for every $\sigma \in \Sigma_R$ . Given assignments $A_L : L \to \Sigma_L$ and $A_R : R \to \Sigma_R$ , the fraction of satisfied edges is $\mathrm{val}_\Psi(A_L, A_R)$ , and

\mathrm{val}(\Psi) \;=\; \max_{A_L, A_R} \mathrm{val}_\Psi(A_L, A_R).

An instance all of whose constraints are $d$ -to-1 is a $d$ -to-1 game.

For $0 < s < c \le 1$ , $\mathrm{Gap\text{-}}d\mathrm{\text{-}to\text{-}}1_r(c,s)$ is the promise problem: given a $d$ -to-1 game with both alphabets of size at most $r$ , distinguish $\mathrm{val}(\Psi) \ge c$ from $\mathrm{val}(\Psi) \le s$ . Writing $\mathrm{GapPLC}_r(c,s)$ for the same promise problem over all projection instances, the PCP theorem together with the parallel repetition theorem gives that $\mathrm{GapPLC}_{r}(1,\varepsilon)$ is NP-hard for a suitable $r = r(\varepsilon)$ (Theorem 1.2 of the source); this mission takes that statement as an external input.

Formalization targets

Goal — Theorem 1.6 of the source

\forall \varepsilon > 0 \ \exists r \in \mathbb{N}^{+} : \quad \mathrm{Gap\text{-}4\text{-}to\text{-}1}_r(1,\varepsilon) \text{ is NP-hard.}

In Lean this is stated as a polynomial-time gap-preserving reduction: for every $\varepsilon > 0$ there is a soundness threshold $s \in (0,1)$ such that for every source alphabet bound $r_0$ there is a target alphabet bound $r$ and a polynomial-time computable map sending projection label-cover instances with alphabets of size at most $r_0$ and value $1$ to $4$ -to-1 games with alphabets of size at most $r$ and value $1$ , and instances of value at most $s$ to $4$ -to-1 games of value at most $\varepsilon$ . Combined with the NP-hardness of $\mathrm{GapPLC}_{r_0}(1,s)$ , this is exactly Theorem 1.6.

Supporting targets

The milestone list follows the source's own numbering: the hardness of approximate colouring of $3$ -uniform hypergraphs that starts the construction (Theorem 3.1), the two Grassmann decoding theorems the inner PCP rests on (Theorems 3.2 and 3.3), the sunflower bound on zoom-outs (Lemma 3.8), and the linear-algebraic layer connecting NAE-satisfying bilinear forms with their tensor decompositions (Propositions 4.13, 4.14 and Corollary 4.15).

Significance

Theorem 1.6 confirms the $4$ -to-1 Games Conjecture, the first of Khot's $d$ -to-1 conjectures to be settled with perfect completeness. Via known reductions it yields: for every $k$ , it is NP-hard to $k$ -colour a $3$ -colourable graph (previously known for $k = 5$ ); for every $\delta>0$ , it is NP-hard to find an independent set of relative size $\delta$ in a $2$ -colourable $3$ -uniform hypergraph; and hardness results for low-rank matrix completion.

None of this material is formalized today. Mathlib has no label cover, no PCP machinery, no Grassmann graph and no complexity classes beyond the computability layer. A complete development therefore contributes reusable infrastructure — finite two-prover games and their value, gap-preserving reductions, the Grassmann graph over $\mathbb{F}_2$ and its agreement tests — well beyond this single theorem.

Difficulty

The obvious attempt is to redo the $2$ -to-1 construction with a perfectly complete outer PCP, namely hardness of systems of quadratic equations over $\mathbb{F}_2$ in place of linear ones. This fails at composition: the Grassmann agreement test, the only known device that produces $d$ -to-1 constraints, is a test for linear functions and cannot certify quadratic constraints. Linearizing the quadratic equations by a low-rank test destroys the covering property of the outer PCP, which is what makes the composed soundness analysis work. The source paper's answer is a three-layer construction (outer, middle and inner PCP) with a lazy parallel repetition in the middle layer and an inner PCP based on a tensor of the standard Grassmann encoding with Golowich's low-rank variant.

Formalization scope

All objects are finite and explicit. A label-cover instance carries left vertices $\{0,\dots,n_L-1\}$ , right vertices $\{0,\dots,n_R-1\}$ , alphabets $\{0,\dots,|\Sigma_L|-1\}$ and $\{0,\dots,|\Sigma_R|-1\}$ , a finite edge set, and a projection map for every pair of vertices; only projection instances are representable, as in Definition 1.1. The value is the supremum over all pairs of assignments of the fraction of satisfied edges, taken in $\mathbb{R}$ ; when there are no edges, or no assignments at all, the convention gives value $0$ . A tripled set (Definition 4.1) is modelled as $\iota \times \{0,1,2\}$ , with the triple indexed by $i$ being $\{(i,0),(i,1),(i,2)\}$ . The Grassmann objects live in $\mathbb{F}_2^n$ modelled as $\mathrm{Fin}\,n \to \mathbb{Z}/2$ , and all probabilities are ratios of cardinalities of finite sets of subspaces, with the convention that an empty denominator gives $0$ .

Hardness is not stated as "NP-hard" — no notion of NP is available — but as the existence of a reduction. This matters: a reduction required only to preserve the gap, with no computability condition, would be trivially satisfiable by a map that inspects the value of its input and returns one of two fixed instances. The formalization therefore requires the reduction map to be computed by a Turing machine within a polynomial time bound, using Mathlib's Turing.TM2ComputableInPolyTime together with an explicit binary encoding of instances. The NP-hardness of the source problem $\mathrm{GapPLC}_{r_0}(1,s)$ (Theorem 1.2, i.e. the PCP theorem plus parallel repetition) is an external input and is not part of this mission.

Contributions of intermediate infrastructure are welcome: the games of Sections 4–6 (Game1a, Game1b, Game2a, Game2b, Game2c, Game3) and their completeness and soundness lemmas are the natural next layer of milestones, as are the covering properties of Appendix C and the list-decoding bounds of Appendix E.

Selected references

Yumou Fei, Dor Minzer, Shuo Wang, On the Hardness of 4-to-1 Games with Perfect Completeness, ECCC TR26-179 (2026), https://eccc.weizmann.ac.il/report/2026/179/
Subhash Khot, On the power of unique 2-prover 1-round games, STOC 2002, https://doi.org/10.1145/509907.509985
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra, Towards a proof of the 2-to-1 games conjecture?, STOC 2018, https://doi.org/10.1145/3188745.3188804
Subhash Khot, Dor Minzer, Muli Safra, Pseudorandom sets in Grassmann graph have near-perfect expansion, FOCS 2018, https://doi.org/10.1109/FOCS.2018.00062
Louis Golowich, New Explicit Constant-Degree Lossless Expanders, FOCS 2023, https://arxiv.org/abs/2306.07551

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