Combinatorics

Motivation

The nearest common ancestor problem asks, for a rooted tree and two of its vertices $x$ and $y$, for the deepest vertex that is an ancestor of both, written $\operatorname{nca}(x,y)$. It appears as a subroutine in string algorithms (suffix trees), in graph algorithms (path queries, dominators) and in the analysis of set-union structures. Aho, Hopcroft and Ullman (On finding lowest common ancestors in trees, SIAM J. Comput. 5, 1976) posed it in several versions, differing in how much the tree changes while the queries are answered.

Harel and Tarjan (Fast Algorithms for Finding Nearest Common Ancestors, SIAM J. Comput. 13, 1984) study how the answer depends on the machine model. On a random-access machine, where addresses can be computed arithmetically, they preprocess a static tree in linear time and then answer each query in constant time. On a pointer machine, where memory can only be traversed by following pointers, their §2 shows that no representation of the tree allows constant-time queries: $\Omega(\log\log n)$ steps are needed in the worst case. This mission formalizes that lower bound.

Timeline.

• 1976: Aho, Hopcroft and Ullman give an $O(\log\log n)$-per-query random-access algorithm for static trees.
• 1976: van Leeuwen (Finding lowest common ancestors in less than logarithmic time, unpublished report, reference [14] of Harel–Tarjan) gives an $O(n + m\log\log n)$ algorithm for static trees that runs on a pointer machine.
• 1984: Harel and Tarjan prove Theorem 1, the matching $\Omega(\log\log n)$ lower bound for pointer machines, and the $O(1)$-per-query random-access algorithm.

Setting

A pointer machine stores its data as a collection of nodes. Each node has a fixed number of fields, and a pointer field holds either a node or nil. The machine can follow a pointer from a node it holds, but it cannot compute an address. Following Harel and Tarjan (p. 340), a static tree is represented by a list structure: each tree vertex $v$ is represented by a single node $\mathrm{rep}(v)$, distinct vertices by distinct nodes, and the structure may contain further nodes that represent no vertex. Each node has two pointer fields; the paper reduces any fixed number of pointers to two "without loss of generality". To answer a query on $x$ and $y$, the machine is given pointers to $\mathrm{rep}(x)$ and $\mathrm{rep}(y)$ and must return a pointer to $\mathrm{rep}(\operatorname{nca}(x,y))$.

The node $b$ is accessible from $a$ in $j$ steps or less if it can be reached from $a$ by following at most $j$ pointers. Write $\mathrm{acc}_j(a)$ for the set of such nodes. A run of $t$ steps from input nodes $a$ and $b$ is a sequence $n_1,\dots,n_t$ in which each $n_s$ is the content of a pointer field of a node among $a, b, n_1, \dots, n_{s-1}$. A query with answer $c$ is answered in $k$ steps if some run of at most $k$ steps holds $c$.

The tree is the complete binary tree $T$ of height $h$, with $n = 2^h$ leaves. Its vertices are the words $w \in \{0,1\}^{\le h}$ (the root-to-vertex path, $0$ = left), the ancestors of $v$ are its prefixes, the depth of $w$ is $|w|$ and its height is $h - |w|$. Then $\operatorname{nca}(x,y)$ is the longest common prefix of $x$ and $y$. Logarithms are binary: $\lg = \log_2$.

Formalization targets

Goal: Theorem 1 in the explicit form of its proof

For every $h$, every node type, every list structure with two pointers per node and every injective representation $\mathrm{rep}$ of the complete binary tree with $n = 2^h$ leaves: if every nca query on two leaves is answered in $k$ steps, then

This is the last display of the proof (p. 341), which is what the paper's $\Omega(\log\log n)$ means. The representation is arbitrary and is quantified before the query bound, so the bound holds for every representation.

Milestones: the claims of the proof

1. A query answered in $k$ steps reaches only nodes in $\mathrm{acc}_k(\mathrm{rep}(x)) \cup \mathrm{acc}_k(\mathrm{rep}(y))$.
2. $|\mathrm{acc}_j(a)| \le 2^{j+1} - 1$ for every node $a$.
3. With $A_x$ the set of vertices whose nodes are accessible from $\mathrm{rep}(x)$ in $k$ steps or less: for a nonleaf $w$ with children $u, v$, either $w \in A_x$ for every leaf $x$ below $u$, or $w \in A_y$ for every leaf $y$ below $v$.
4. A vertex of height $i \ge 1$ lies in $A_x$ for at least $2^{i-1}$ leaves $x$.

where $L$ is the set of leaves.

Significance

The result. Theorem 1 shows that van Leeuwen's pointer-machine algorithm for static trees is optimal up to a constant factor, and that the constant-time queries of the paper's §§3–5 depend on address arithmetic. It is an early nontrivial lower bound for pointer machines on a natural problem; the paper compares it with Tarjan's lower bound for disjoint-set union on a pointer machine (J. Comput. System Sci. 18, 1979).

Formalizing it. The theorem is proved in the paper; as far as could be determined no machine-checked version exists, and Mathlib has no pointer-machine model. The mission produces an explicit, reusable definition of pointer-machine runs and accessibility together with a complete proof of the explicit bound. A formal model of this kind is the precondition for stating any other pointer-machine lower bound.

Difficulty

The statement must hold for every representation, including structures with many auxiliary nodes and arbitrary pointers between tree nodes. Arguing about one natural representation, such as parent pointers, where a leaf is far from its ancestors, says nothing about other representations: a structure with shortcut pointers or auxiliary nodes may bring some ancestors close to some leaves, and the bound must survive every such choice. In the formal setting the counting also has to handle overlaps: nodes reachable from several leaves, nodes that represent no vertex, and pointer cycles.

Formalization scope

• Model. Nodes form an arbitrary type N, not necessarily finite. ptr : N → Fin 2 → Option N gives the two pointer fields (none = nil), and rep : Vertex h → N is required to be injective. acc ptr j a is defined recursively. Run ptr a b t held is an inductive predicate for runs of $t$ steps, AnsweredIn asks for some run of at most $k$ steps holding the answer, and AnswersLeafQueriesIn ptr rep k requires this for every pair of leaves.
• Conventions.
  • Two pointer fields per node, as the paper's "without loss of generality" reduction allows; the reduction itself is not formalized.
  • Only queries on two leaves are assumed answerable. This is weaker than all queries, so the theorem is at least as strong as the paper's.
  • Time is counted as pointer-following steps. Mutation of the structure during a query and non-pointer fields are not modelled: neither lets the machine hold a node it has not reached by following pointers. The clause "the algorithm remembers nothing between queries" is built into the static structure.
  • Vertex h is {s : List Bool // s.length ≤ h}, nca is the longest common prefix, and a separate theorem identifies it with the Appendix's deepest common ancestor. $n = 2^h$ counts leaves, not vertices.
  • $\lg$ is Real.logb 2. For $h = 0$ Lean's $\log_2 0 = 0$ gives the true statement $k > -2$; for $h \ge 1$, $\lg\lg n = \log_2 h$.
  • Cardinalities in the milestones are Set.encard in $\mathbb N \cup \{\infty\}$, so finiteness is part of each claim. Divisions are cleared: $h\,2^h \le 2\sum_x |A_x|$.
• Ruling out trivial formalizations. The hypothesis AnswersLeafQueriesIn is satisfiable: the parent-pointer representation answers every leaf query in $h$ steps. If rep were not injective, a constant rep would answer every query in zero steps, so injectivity is kept in the goal. The milestones do not need it and do not assume it.
• Infrastructure. The goal needs finite-set counting over the leaves of the complete binary tree and a double count over heights. The run and accessibility definitions are reusable for other pointer-machine arguments. Proofs of the milestones, and of the ℕ form $h < 2^{k+2}$ that the goal reduces to, are welcome.

Selected references

• D. Harel, R. E. Tarjan, Fast Algorithms for Finding Nearest Common Ancestors, SIAM J. Comput. 13(2):338–355, 1984. https://doi.org/10.1137/0213024
• A. V. Aho, J. E. Hopcroft, J. D. Ullman, On finding lowest common ancestors in trees, SIAM J. Comput. 5(1):115–132, 1976. https://doi.org/10.1137/0205011
• A. Schönhage, Storage modification machines, SIAM J. Comput. 9(3):490–508, 1980. https://doi.org/10.1137/0209036
• R. E. Tarjan, A class of algorithms which require nonlinear time to maintain disjoint sets, J. Comput. System Sci. 18(2):110–127, 1979. https://doi.org/10.1016/0022-0000(79)90042-4

Why the assignment dual matters

The subgradient method maximizes a concave, piecewise-linear function $w(\pi)=\min_k\{c_k+\pi\cdot v_k\}$ by moving along a subgradient $v_k$ of an active piece with a prescribed step. Held, Wolfe and Crowder's 1974 paper Validation of subgradient optimization tested the method on three families of Lagrangean duals from combinatorial optimization — the assignment problem, a relaxation of the travelling salesman problem in the style of Held and Karp, and multicommodity flows — and gave the first systematic account of when the method works in practice.

On randomly generated assignment problems of order $n\le 30$ the authors observed that the method usually did not merely converge: it stopped, after finitely many steps, at an iterate whose subgradient was exactly zero. Their explanation is a structural fact about the assignment dual, Theorem 3.1 of the paper: when the optimal assignment is unique — the typical case for random integer costs — the set of optimal dual prices has full dimension $n$, so a sequence of steps of decreasing length can land inside it. This mission formalizes that theorem and the steps of its proof.

Setting

There are $n$ men and $n$ jobs, and a real $n\times n$ cost matrix $A=(a_{ir})$: $a_{ir}$ is the cost for which man $i$ does job $r$. A one-to-one assignment is a permutation $\sigma$ of $\{1,\dots,n\}$, where $\sigma(r)$ is the man doing job $r$; its cost is $\sum_r a_{\sigma(r)\,r}$. The assignment problem (3.1) asks for a permutation of minimal cost; the assignment is unique if exactly one permutation attains that minimum.

The linear relaxation of (3.1), over doubly stochastic matrices $x=(x_{ir})$, has the dual linear program (3.2), $\max\{\sum_i\pi_i+\sum_r\rho_r : \pi_i+\rho_r\le a_{ir}\}$. For fixed prices $\pi\in\mathbb R^n$ on the men the best $\rho$ is $\rho_r=\min_s[a_{sr}-\pi_s]$, which leaves the dual function (3.3)

the inner minimum being over the men $s$ for each job $r$. The optimal set is $\Omega=\{\pi : w(\pi')\le w(\pi)\ \text{for all }\pi'\}$.

To put $w$ in the form $\min_k\{c_k+\pi\cdot v_k\}$ the paper uses assignments in a weaker sense: arbitrary functions $A:\{1,\dots,n\}\to\{1,\dots,n\}$, $n^n$ of them, with cost $c_A=\sum_r a_{A(r)\,r}$ and vector $(v_A)_i=1-\#\{r:A(r)=i\}$ (3.4). The subgradient step raises the price of a man assigned no job and lowers the price of a man assigned several; $v_A=0$ exactly when $A$ is a permutation.

In the Lean development these are assignCost, assignVec, IsOptimalAssignment, w and optSet in the namespace HeldWolfeCrowder.Assignment.

Formalization targets

Goal: Theorem 3.1 (p. 70)

If the assignment problem has a unique optimal permutation, then

The hypothesis is uniqueness among permutations; the conclusion is the dimension of the affine hull of the optimal set.

Milestones, in the order the proof uses them

1. Eq. (3.4): $w(\pi)=\min_A\{c_A+\sum_i\pi_i(v_A)_i\}$ over all $n^n$ assignments $A$.
2. §3, Eqs. (3.1)–(3.3): $w$ attains its maximum, and $\max w$ equals the cost of an optimal permutation.
3. Eq. (3.5): if $\sigma$ is the unique optimal permutation, some maximizer $\bar\pi$ of $w$ has, for every job $r$, the minimum $\min_s[a_{sr}-\bar\pi_s]$ attained only at $s=\sigma(r)$.
4. Eq. (3.6): for an optimal permutation $\sigma$, the set $\Pi=\{\pi : a_{ir}-\pi_i>a_{\sigma(r)\,r}-\pi_{\sigma(r)}\ \text{for all } r,\ i\ne\sigma(r)\}$ is convex and open, $v=0$ on it, and $\Pi\subseteq\Omega$.

Significance

The theorem turns an empirical observation into a statement about the problem: finite termination of the subgradient method on assignment problems is a property of the dual, not luck. Since $w$ is unchanged by adding the same constant to every price, $\Omega$ always contains a line; Theorem 3.1 says that, under uniqueness, it is as large as it can be. The paper (p. 70) cites the argument of its Section 2 that, with a full-dimensional optimal set, termination of the method is "nearly certain".

The result is proved in the paper; none of it is known to be machine-checked. What the formalization adds is a checked link between three classical ingredients: the integrality of the assignment polytope (Birkhoff–von Neumann, which Mathlib has as doublyStochastic_eq_convexHull_permMatrix), linear-programming duality, and strict complementary slackness, which neither Mathlib nor the platform has in the form needed. The piecewise-linear representation (3.4) is reusable wherever the assignment dual appears as a Lagrangean subproblem.

Difficulty

The inclusion $\Pi\subseteq\Omega$ is elementary; the substance is that $\Pi$ is nonempty. The obvious candidate — any optimal dual solution — fails: an optimal $\pi$ may leave ties $a_{sr}-\pi_s=a_{\sigma(r)\,r}-\pi_{\sigma(r)}$ for some $s\ne\sigma(r)$, so it sits on the boundary of $\Omega$ and shows nothing about dimension. What is needed is an optimal price vector with all these inequalities strict at once, and uniqueness of the optimal permutation is a statement about the primal side only; transferring it to the dual side goes through the linear relaxation (3.1), whose uniqueness is not the hypothesis, and through a strict complementarity property that is not available in Mathlib or on the platform.

Formalization scope

Men and jobs are both Fin n; the costs are a : Matrix (Fin n) (Fin n) ℝ with a i r the cost of man i on job r; prices are π : Fin n → ℝ (no inner product or norm is needed, so no EuclideanSpace). The inner minimum of (3.3) is Finset.univ.inf' over the men, well defined for every n. One-to-one assignments are Equiv.Perm (Fin n) with σ r the man doing job r, so the orientation of the matrix matches (3.3); arbitrary assignments are functions Fin n → Fin n. "Of dimension $n$" is Module.finrank ℝ (vectorSpan ℝ (optSet a)) = n. The page prints the index condition of (3.6) as "$i\ne r$"; the formalization uses $i\ne\sigma(r)$, which is what the argument requires. The case $n=0$ is allowed and trivial.

A statement asserting only that $\Omega$ is nonempty, or that it has dimension at least one, is not this theorem: both hold for every cost matrix, the second because $\Omega$ is invariant under adding a constant to all prices. The goal requires the full value $n$, and its hypothesis is uniqueness of the optimal permutation, not of the optimal linear-programming solution.

A complete development needs: the assignment linear program and its integrality (Mathlib's Birkhoff–von Neumann theorem), weak and strong duality between (3.1) and (3.2) or directly $\max w=\min_\sigma c_\sigma$, and a strict complementarity statement for this primal–dual pair; the last two are reusable beyond this mission. Contributions of any of these, and of alternative arguments for (3.5) that avoid strict complementary slackness, are welcome.

Selected references

• M. Held, P. Wolfe, H. P. Crowder, Validation of subgradient optimization, Mathematical Programming 6 (1974) 62–88. https://doi.org/10.1007/BF01580223
• M. Held, R. M. Karp, The traveling-salesman problem and minimum spanning trees: Part II, Mathematical Programming 1 (1971) 6–25. https://doi.org/10.1007/BF01584070
• H. W. Kuhn, The Hungarian method for the assignment problem, Naval Research Logistics Quarterly 2 (1955) 83–97. https://doi.org/10.1002/nav.3800020109
• A. J. Goldman, A. W. Tucker, Theory of linear programming, in H. W. Kuhn, A. W. Tucker (eds.), Linear Inequalities and Related Systems, Annals of Mathematics Studies 38, Princeton University Press, 1956, 53–97.
• Mathlib, Mathlib/Analysis/Convex/Birkhoff.lean (Birkhoff–von Neumann theorem, doublyStochastic_eq_convexHull_permMatrix). https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Analysis/Convex/Birkhoff.lean

Motivation

Integer programs are routinely solved by first solving their linear programming (LP) relaxation and then searching for an integer optimum near the fractional one. How near an integer optimum must be is the subject of proximity theorems. They bound the search region of branch-and-bound and of dynamic programming, and they turn a fractional optimum into a starting point for exact algorithms.

The classical bound is due to Cook, Gerards, Schrijver and Tardos (Math. Programming 34, 1986): for an integer program in inequality form $\max\{c^Tx : Ax\le b,\ x\in\mathbb Z^n\}$ that is feasible and bounded, every optimal LP solution $x^*$ has an optimal integer solution $z^*$ with $\|x^*-z^*\|_\infty\le n\cdot\delta$, where $\delta$ is the largest absolute value of a subdeterminant of $A$. For programs in standard form $Ax=b$ with $m$ rows this gives, via the Hadamard bound, $\|z^*-x^*\|_1\le n^2\cdot m^{m/2}\Delta^m$, which grows with the number of variables $n$.

Eisenbrand and Weismantel (ACM Trans. Algorithms 16(1), Article 5, 2019; conference version SODA 2018) removed the dependence on $n$ altogether, using the Steinitz lemma on rearranging vectors so that all partial sums stay short. Their bound depends only on $m$ and on the largest absolute value $\Delta$ of an entry of $A$, and it is the basis of their faster algorithms for integer programs with few constraints.

Setting

Fix natural numbers $m$ (rows) and $n$ (variables). The data are a matrix $A\in\mathbb Z^{m\times n}$, a right-hand side $b\in\mathbb Z^m$, an objective $c\in\mathbb Z^n$ and upper bounds $u\in\mathbb N^n$. A natural number $\Delta$ bounds the entries: $|a_{ij}|\le\Delta$ for all $i,j$. The integer program (10) is

and its LP relaxation is the same problem over $x\in\mathbb R^n$. Its feasible region $P=\{x\in\mathbb R^n: Ax=b,\ 0\le x\le u\}$ is a polytope, lpPolytope A b u. An optimal vertex solution is an optimal solution of the LP relaxation (IsLPOptimal) that is an extreme point of $P$. An optimal integer solution is IsIPOptimal. Both are maxima.

Distances are measured in the $\ell_1$-norm $\|z-x\|_1=\sum_i|z_i-x_i|$.

A vector $y\in\mathbb Z^n$ is a cycle of $z^*-x^*$ (Eq. (14)) if $Ay=0$ and, for every $i$, $|y_i|\le|(z^*-x^*)_i|$ and $y_i(z^*-x^*)_i\ge0$: an integer kernel vector that is sign-compatible with $z^*-x^*$ and dominated by it (IsCycle).

The Steinitz lemma (Theorem 1.1) concerns vectors $x_1,\dots,x_n$ in an $m$-dimensional normed space with $\sum_i x_i=0$ and $\|x_i\|\le1$. It asserts a permutation $\pi$ with $\|\sum_{j\le k}x_{\pi(j)}\|\le c(m)$ for all $k$, and the paper uses Sevast'anov's constant $c(m)=m$.

Formalization targets

Goal: Theorem 3.3 (p. 5:8)

If (10) has an integer feasible point and $x^*$ is an optimal vertex solution of its LP relaxation, then there is an optimal solution $z^*$ of (10) with

The constant is the paper's. The goal holds for all $m$, $n$, $b$, $c$ and $u$; only $m$ and $\Delta$ enter the bound.

Milestones, in the order the proof uses them

1. Lemma 3.1 (p. 5:8): for an LP optimum $x^*$, an integer optimum $z^*$ and a cycle $y$ of $z^*-x^*$, the vector $z^*-y$ is integer feasible, $x^*+y$ is LP feasible, and $c^Ty\le0$.
2. Lemma 3.2 (p. 5:8): if $z^*$ minimizes $\|z^*-x^*\|_1$ among the optimal integer solutions, then $z^*-x^*$ has no nonzero cycle.
3. Theorem 1.1 with $c(m)=m$ (p. 5:4): the Steinitz lemma in any $m$-dimensional real normed space.
4. Proof of Theorem 3.3 (pp. 5:8–5:9): round a vertex $x^*$ towards an integer vector and write $\{x^*\}$ for the remainder. Then $\|-A\{x^*\}\|_\infty\le\Delta m$ and $-A\{x^*\}=w_1+\dots+w_m$ with integer $w_j$, $\|w_j\|_\infty\le\Delta$.
5. Proof of Theorem 3.3, Eq. (20) (p. 5:9): a sequence of integer vectors of $\ell_\infty$-norm at most $m\Delta$ in which no value repeats $m+1$ times has length at most $m(2m\Delta+1)^m$.
6. Eq. (21) (p. 5:9), a consequence: $c^T(x^*-z^*)\le\|c\|_\infty\cdot m(2m\Delta+1)^m$ for every optimal integer solution $z^*$.

Significance

The bound is independent of the number of variables. Combined with the paper's dynamic program, it gives the paper's running-time results for integer programs with upper bounds: an optimal LP vertex is computed, and the integer optimum is searched for within an $\ell_1$-ball of radius $m(2m\Delta+1)^m$ around it. Eq. (21) bounds the absolute integrality gap by the same quantity, scaled by $\|c\|_\infty$. The Steinitz lemma with constant $m$ is a general tool in discrepancy theory and in scheduling algorithms.

All of these results have published proofs. No machine-checked proof of Theorem 3.3 or of the Steinitz lemma is known to this mission, and Mathlib has no Steinitz lemma. The mission asks for complete Lean proofs of the milestones and of the goal. A proof of the Steinitz lemma with constant $m$ for arbitrary norms is reusable well beyond integer programming.

Difficulty

Lemmas 3.1 and 3.2 and the counting step are elementary. The substance lies in two places. The first is the Steinitz lemma with the linear constant $m$ for an arbitrary norm: the bound must hold uniformly in the number $n$ of vectors, and the constant must be exactly $m$, because the goal's constant $(2m\Delta+1)^m$ counts integer points of $\ell_\infty$-norm at most $m\Delta$. The second is the passage from a vertex to at most $m$ fractional coordinates. The paper argues this in one sentence ("$x^*$ has at most $m$ positive entries"), which is not literally true for (10) with upper bounds: coordinates at their upper bound $u_i>0$ are positive. The correct fact concerns coordinates strictly between $0$ and $u_i$, and it has to be derived from the extreme-point property of $P$.

Formalization scope

• All declarations live in the namespace IPProximity.Eisenbrand. The data are integral: A : Matrix (Fin m) (Fin n) ℤ, b : Fin m → ℤ, c : Fin n → ℤ, u : Fin n → ℕ (entries $u_i=0$ allowed), Δ : ℕ. They are cast to ℝ once, inside the LP definitions. $m=0$ and $n=0$ are allowed.
• "Vertex" is Mathlib's Set.extremePoints ℝ (lpPolytope A b u). It is not defined through bases or by counting fractional coordinates.
• The $\ell_1$-distance is the explicit sum ∑ i, |(z i : ℝ) - x i|. Mathlib's norm on Fin n → ℝ is the sup norm, and it is used only where the paper has $\|\cdot\|_\infty$ (the $\|c\|_\infty$ of Eq. (21)).
• The goal adds one hypothesis the paper leaves implicit: (10) has an integer feasible point. The paper's proof begins with "Let $z^*$ be an optimal integer solution"; without this hypothesis the conclusion is false.
• Eq. (14) is formalized literally, so $y=0$ is a cycle, and Lemma 3.2 is stated for nonzero cycles, which is what its proof establishes. Dropping the vertex hypothesis would make the goal false, so the goal keeps it. The constant is exactly $m(2m\Delta+1)^m$, with no hidden existential constant.
• The Steinitz milestone is stated for any finite-dimensional real normed space of dimension $m$ with the explicit constant $m$. The goal needs only the $\ell_\infty$ case on $\mathbb R^m$.
• Out of scope: the dynamic program and the running-time theorems of Sections 2 and 4, and the refinement $\|z^*-x^*\|_1\le2\Delta$ for $m=1$.

Contributions welcome: proofs of any milestone, in particular the Steinitz lemma, and a proof of the goal from the milestones.

Selected references

• F. Eisenbrand, R. Weismantel, Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma, ACM Transactions on Algorithms 16(1), Article 5, 2019. https://doi.org/10.1145/3340322
• W. Cook, A. M. H. Gerards, A. Schrijver, É. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming 34, 251–264, 1986. https://doi.org/10.1007/BF01582230
• E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme, Journal für die reine und angewandte Mathematik 143, 128–176, 1913. https://doi.org/10.1515/crll.1913.143.128
• S. Sevast'janov, Approximate solution of some problems of scheduling theory (in Russian), Metody Diskretnogo Analiza 32, 66–75, 1978 (reference [31] of the paper).
• V. S. Grinberg, S. V. Sevast'yanov, Value of the Steinitz constant, Functional Analysis and Its Applications 14(2), 125–126, 1980 (reference [16] of the paper).

Motivation

The maximum flow problem asks how much of a commodity can be sent from a source to a sink through a network whose arcs have capacities. It underlies bipartite matching, transportation, scheduling and many reductions in combinatorial optimization. The classical method for it, the labeling method of Ford and Fulkerson (Flows in Networks, 1962), repeatedly finds an augmenting path and pushes flow along it. With integer capacities it terminates, but the number of augmentations can be as large as the maximum flow value itself, and Edmonds and Karp exhibit a four-node network on which this happens (p. 250). With irrational capacities the method need not terminate at all.

Edmonds and Karp, Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems, J. ACM 19(2):248–264, 1972 (doi:10.1145/321694.321699), showed that two simple rules for choosing the augmenting path repair this. The first, augmenting along a path with fewest arcs, is the subject of mission 1 of this series. This mission covers the second (§1.3): augment along a path that gives the largest possible augmentation. For integer capacities the number of augmentations then grows only logarithmically in the maximum flow value.

Setting

A network $N$ has a finite set $V$ of nodes, a source $s$ and a sink $t \neq s$, and a set of arcs, ordered pairs $(u,v)$ with $u \neq v$, at most one from each node to another. One arc is the return arc $(t,s)$; the other arcs form the set $A$, and each $(u,v) \in A$ has a capacity $c(u,v) > 0$. A flow is a nonnegative function $f$ on the arcs of $N$ with $f(u,v) \le c(u,v)$ on $A$ and flow conservation at every node, $s$ and $t$ included. Its value is $f(t,s)$, the flow returned along the return arc; a maximum flow has the largest value among all flows, and $f^*(t,s)$ denotes that value.

The residual network $N^f$ has an arc $(u,v)$ whenever $(u,v) \in A$ and $c(u,v) - f(u,v) > 0$, or $(v,u) \in A$ and $f(v,u) > 0$. An augmenting path is a directed path $s = u_1, \dots, u_p = t$ of distinct nodes in $N^f$. Each of its arcs $(u,v)$ has a residual amount $e(u,v)$, equal to $c(u,v) - f(u,v)$, $f(v,u)$, or $c(u,v) - f(u,v) + f(v,u)$ according to which of $(u,v)$, $(v,u)$ lie in $A$, and the path's augmentation is $\varepsilon = \min e(u_i, u_{i+1})$. Augmenting increases $f(t,s)$ by $\varepsilon$ and changes the flow on the arcs of the path accordingly, with the paper's own rule when both $(u,v)$ and $(v,u)$ are arcs. The labeling method produces flows $f^0, f^1, \dots$ by augmenting along a path relative to $f^k$ as long as one exists.

The rule studied here chooses, at every step, an augmenting path whose $\varepsilon$ is at least that of every other augmenting path relative to the current flow. The bound involves an integer $M > 1$ such that every partition of the nodes into $X \ni s$ and $\bar X \ni t$ has at most $M$ arcs of $N$ with one end on each side.

Formalization targets

Goal: Theorem 2 (p. 253)

For a network with integer capacities, $M$ as above, and a run $f^0, \dots, f^K$ of the labeling method with maximum augmentations started from an integer-valued flow,

and if no augmenting path relative to $f^K$ exists, then $f^K$ is a maximum flow.

Milestones

The milestone list follows the paper's argument:

1. augmentation produces a flow of value $f(t,s) + \varepsilon$ (§1.1, p. 249);
2. a flow is maximum if and only if it has no augmenting path (§1.1, pp. 249–250);
3. with integer capacities, $\varepsilon$ is a positive integer and the flows of the method stay integer-valued (§1.1, p. 250);
4. the cut inequality $c(X,\bar X) \ge f(X,\bar X) - f(\bar X,X) = f(t,s)$ (p. 254);
5. $f^*(t,s) - f^k(t,s) \le \varepsilon^k M$, where $\varepsilon^k = f^{k+1}(t,s) - f^k(t,s)$ (p. 254);
6. $f^*(t,s) - f^{k+1}(t,s) \le [f^*(t,s) - f^k(t,s)](1 - M^{-1})$ (p. 254);
7. $f^*(t,s) - f^k(t,s) \le f^*(t,s)(1 - M^{-1})^k$ (p. 254).

Significance

Theorem 2 was among the first bounds showing that a maximum flow algorithm can be made polynomial in the size of the numbers rather than in their values: since $M \le n^2/2$ and $f^*(t,s)$ is at most $n^2$ times the average capacity, the bound is $O(n^2 \log(n^2 \bar c))$ in terms of the number of nodes $n$ and the average capacity $\bar c$ (p. 254). The largest-augmentation rule, often called the fattest-path or maximum-capacity augmenting path rule, is a standard textbook variant, and its geometric-decrease argument is the model for later capacity-scaling methods, including the scaling algorithm for the Hitchcock problem in §2 of the same paper (mission 3 of this series).

The theorem has been proved since 1972 and appears in standard texts. As far as a platform search shows (2026-09-26), no machine-checked proof of it exists on Prove2Me. The platform does contain LinearOptimization.max_flow_min_cut and LinearOptimization.max_flow_ford_fulkerson_integer_termination, which state max-flow min-cut and termination of the generic method in a different network model (parallel arcs, extended nonnegative capacities, no return arc); they give no count of augmentations and are related work only. This mission would contribute a formal proof of the counting bound together with the general labeling-method facts (milestones 1–3), which mission 1 needs as well.

Difficulty

The obvious argument, that each augmentation raises the value by at least 1, gives only the bound $f^*(t,s)$, and on the four-node example of p. 250 that bound is attained by an arbitrary choice of paths. The logarithmic bound needs a lower bound on the size of the largest augmentation in terms of the remaining gap $f^*(t,s) - f^k(t,s)$. The largest augmentation is defined by comparison with all augmenting paths relative to the current flow, while the gap is a global quantity of the network, and neither integrality nor the maximum-augmentation rule alone controls it. Milestone 2's converse, that a non-maximum flow always admits an augmenting path, is itself the max-flow min-cut theorem in this model, and the formal proof has to establish it for the paper's return-arc model rather than import it from a different one.

Formalization scope

• Nodes form a finite type V with decidable equality. A : Finset (V × V) contains no loops and not $(t,s)$. Capacities are real, c : V → V → ℝ, positive on A. Integrality is the hypothesis IntegralCaps N, and for the initial flow IsIntegralOn N (f 0) (integer values on the arcs of $N$, the return arc included).
• Flows are functions V → V → ℝ constrained only on the arcs of $N$. A maximum flow is the predicate IsMaxFlow, comparing $f(t,s)$ with every flow, not a supremum. The goal takes a maximum flow g as a hypothesis and sets $f^*(t,s) = g(t,s)$; every network has one.
• Augmenting paths are duplicate-free node lists whose consecutive pairs are arcs of $N^f$. The page prints Case (b) of the definition of $\varepsilon_i$ with the same hypothesis as Case (c); the corrected Case (b), $(u,v) \notin A$ and $(v,u) \in A$, is used, as the definition of $N^f$ (p. 251) and the list for $e(u,v)$ (p. 253) confirm.
• A run is IsMaxAugRun N K f P. Its initial flow is arbitrary except for integrality, and each later flow is the augmentation of the previous one along a path of maximum $\varepsilon$ among all augmenting paths.
• The crossing bound CrossArcsBounded N M counts the arcs of $N$, return arc included, with one end on each side of every $s$–$t$ partition. This is the literal reading of p. 253.
• Explicit constants. The bound is exactly $1 + \log_{M/(M-1)} f^*(t,s)$, written (K : ℝ) ≤ 1 + Real.logb ((M : ℝ) / ((M : ℝ) - 1)) (g N.t N.s) with $M > 1$ a natural number. When $f^*(t,s) = 0$, Real.logb gives $0$ and the bound reads $K \le 1$. The contraction factor is 1 - (M : ℝ)⁻¹.
• A statement that bounds only runs of an unsatisfiable step predicate, drops the integrality of $f^0$ or of the capacities (the bound is false without them), or compares $\varepsilon$ only among paths of some restricted class does not formalize Theorem 2. A sorry-free check exhibits a four-node network with integer capacities and a valid maximum-augmentation step.
• Reusable beyond this mission: the return-arc network model, the augmentation step with the paper's opposite-arc rule, the integrality lemma, and the cut inequality. Proofs of any milestone are welcome, as are proofs of the converse in milestone 2 that could later be shared with mission 1.

Selected references

• J. Edmonds, R. M. Karp, Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems, Journal of the ACM 19(2):248–264, 1972. https://doi.org/10.1145/321694.321699
• L. R. Ford, D. R. Fulkerson, Flows in Networks, RAND report R-375-PR, 1962; Princeton University Press, 1962. https://www.rand.org/pubs/reports/R375.html

Motivation

The Steiner problem in graphs asks for the cheapest way to connect a prescribed set of nodes of a network, where intermediate nodes may be used freely. It is the network version of the classical Euclidean Steiner tree problem surveyed by Gilbert and Pollak (SIAM J. Appl. Math. 16, 1968), and it arises wherever a few sites must be joined through an existing network at minimum total cost: communication and pipeline layout, VLSI routing, and phylogenetics. With two terminals it is the shortest-path problem; with all nodes as terminals it is the minimum spanning tree problem; in between it is NP-hard.

Dreyfus and Wagner (Networks 1(3):195–207, 1971) gave the first exact algorithm whose running time is exponential only in the number $k$ of terminals and polynomial in the number $n$ of nodes. The paper states it, as Algorithm A, together with its proof of correctness and an exact count of its elementary operations.

Timeline. 1968: Gilbert and Pollak survey Steiner minimal trees. 1971: Dreyfus and Wagner, a dynamic program over subsets of terminals running in time proportional to $n^3/2 + n^2(2^{k-1}-k-1) + n(3^{k-1}-2^k+3)/2$. 1987: Erickson, Monma and Veinott give the same subset recursion for general network flow problems. 2007: Björklund, Husfeldt, Kaski and Koivisto (STOC 2007) improve the exponential dependence on $k$ for small integer weights. The Dreyfus–Wagner recursion remains the standard exact method and the basis of the fixed-parameter tractability of the problem in $k$.

Setting

A graph $G = (N, A)$ has a finite set $N$ of nodes and a set $A$ of undirected arcs, each arc $a$ having a positive length $|a|$; $G$ is connected. For a set $S \subseteq A$ of arcs, $|S| = \sum_{s \in S} |s|$. A set $S$ connects a node set $X$ if all members of $X$ are joined by paths composed only of arcs in $S$.

Given $Y \subseteq N$, a Steiner path (or Steiner tree) connecting $Y$ is a set $S \subseteq A$ that connects $Y$ with $|S|$ minimum. Its length is the Steiner length $\operatorname{St}(Y)$. For nodes $i, j$, $D(i,j)$ is the length of a shortest path from $i$ to $j$; $D(i,j) = \operatorname{St}(\{i,j\})$.

Algorithm A fixes a linear order of $N$ (so that each nonempty set $D$ has a first element $D[1]$), picks $q \in Y$, sets $C = Y - \{q\}$, and fills a table $S[D, I]$ for nonempty $D \subsetneq C$ and $I \in N$:

and returns

A minimum over an empty set is $+\infty$. In the Lean development these objects are steinerLength, pathDist, tableA and algorithmA in the namespace DreyfusWagner.Steiner.

Formalization targets

Goal: Algorithm A is exact

For every finite connected graph with positive arc lengths, every linear order on its nodes, every $Y$ with $\|Y\| \ge 3$ and every $q \in Y$,

This is the caption of Algorithm A ("Computes the length of the Steiner tree connecting $Y$", p. 203). The statement is an equality, not a bound.

Milestones

In the order the proof uses them:

1. A Steiner path is a tree (§1, p. 197): a minimum connecting arc set contains no cycle.
2. The two-node case (Appendix A, p. 205): $\operatorname{St}(\{i,j\}) = D(i,j)$.
3. Theorem 1 (Appendix A, p. 206): for a Steiner tree $S$, a node $x$ on it, and a set $C$ of arcs of $S$ at $x$, the arcs of $S$ connecting $x$ to the terminals reached through $C$ form a Steiner tree for those terminals together with $x$.
4. Optimal Decomposition Theorem (Appendix A, p. 206): if $\|Y\| \ge 3$ and $q \in Y$, a Steiner tree for $Y$ splits into three disjoint Steiner paths, for $\{p,q\}$, $\{p\} \cup D$ and $\{p\} \cup (Y - D - \{q\})$, where $p \in N$ and $\emptyset \ne D \subsetneq Y - \{q\}$.
5. The recurrence (§2, pp. 199–200): for $\|D\| \ge 2$ and any node $m$,

6. The table invariant (§2, p. 200): $S[D, I] = \operatorname{St}(\{I\} \cup D)$ for every nonempty $D$ and every $I$.

Two companion items accompany the goal: the numerical illustration of §3 (seven nodes, $\operatorname{St}(Y) = 5$, Algorithm A returns $5$), and the exact count of elementary statements of §5, $n^2(2^{k-1}-k-1) + n(3^{k-1}-2^k+3)/2$.