The OR Formalization Drive

Motivation

Regularization — adding a penalty on model complexity to an empirical risk minimization objective — is one of the oldest and most reliably effective tools in statistical learning: ridge regression, LASSO, and margin-based support vector machines all fit this template. For decades the regularization weight and penalty function were chosen heuristically or by cross-validation, with only asymptotic or worst-case generalization bounds explaining why they help. Kuhn, Mohajerin Esfahani, Nguyen & Shafieezadeh-Abadeh's 2019 INFORMS TutORials chapter on Wasserstein distributionally robust optimization (DRO) gives regularization a different, non-asymptotic justification: a decision rule that is robust to adversarial perturbations of its training data, measured in Wasserstein distance, is exactly a regularized empirical risk minimizer — no approximation, no asymptotics. This mission formalizes the general form of that equivalence, Theorem 10 (p. 14), which underlies every regularization-by-robustification result the chapter derives for classification and regression alike.

Setting

Fix a normed space $\Xi = \mathbb{R}^m$, a convex loss function $\ell : \Xi \to \mathbb{R}$, a radius $\varepsilon \ge 0$, and $N$ training samples $\hat\xi_1,\dots,\hat\xi_N \in \Xi$ ($N \ge 1$). The empirical distribution is $\hat P_N = \frac{1}{N}\sum_{i=1}^N \delta_{\hat\xi_i}$. The worst-case risk of $\ell$ at radius $\varepsilon$ (eq. (6), p. 6) is

$$R_{\varepsilon,p}(\hat P_N,\ell) = \sup_{Q \in B_{\varepsilon,p}(\hat P_N)} E_Q[\ell(\xi)],$$

where $B_{\varepsilon,p}(\hat P_N)$ is the type-$p$ Wasserstein ball of radius $\varepsilon$ around $\hat P_N$ (Definition 1, p. 3) — the set of probability measures on $\Xi$ within Wasserstein distance $\varepsilon$ of the empirical distribution, under a norm $\|\cdot\|$ on $\Xi$ and its transportation exponent $p$. The Lipschitz modulus $\mathrm{Lip}(\ell) = \sup_{\xi\ne\xi'} \frac{|\ell(\xi)-\ell(\xi')|}{\|\xi-\xi'\|}$ (possibly $+\infty$) measures how fast $\ell$ can grow.

Formalization targets

Goal (Theorem 10, convex loss and $p=1$). If $\ell$ is convex and $p=1$, then the worst-case risk of $\ell$ over the type-1 Wasserstein ball around the empirical distribution coincides with the Lipschitz-regularized empirical loss:

$$R_{\varepsilon,1}(\hat P_N,\ell) = R(\hat P_N,\ell) + \varepsilon \cdot \mathrm{Lip}(\ell),$$

where $R(\hat P_N,\ell) = \frac{1}{N}\sum_{i=1}^N \ell(\hat\xi_i)$ is the ordinary empirical risk. The left side is the worst-case expected loss under adversarial perturbations of the training data of bounded aggregate Wasserstein cost; the right side is the ordinary empirical risk plus a regularization term proportional to the perturbation radius $\varepsilon$ and the loss's own Lipschitz modulus — no approximation, an exact equality for every convex $\ell$.

Significance

This is the single result underlying every specific regularization-by-robustification corollary the chapter derives — the $\ell_2$-regularized SVM and its $\ell_1$/$\ell_\infty$ variants, regularized logistic regression, and the regression analogue with a Lipschitz loss (Proposition 3) — because each of those is obtained by substituting a specific convex, Lipschitz loss (hinge, logloss, a linear-model margin loss) for the general $\ell$ here and reading off $\mathrm{Lip}(\ell)$ in closed form. Theorem 10 is exact ($p=1$, Ξ = R^m) precisely because a type-1 transportation cost is dual to the Lipschitz modulus (a consequence of the Kantorovich- Rubinstein duality this book's earlier duality theorems establish), whereas the corresponding statement for $p \ge 2$ (Theorem 11 elsewhere in the chapter) needs a strictly stronger hypothesis on $\ell$. Formalizing Theorem 10 fixes, machine-checkably, the exact scope of the equivalence — which losses qualify (convex, no boundedness or smoothness needed), which transportation exponent is required ($p=1$, not any $p\ge1$), and that the equality is exact, not an upper or lower bound — the single fact every classification- and regression-specific robustification corollary in the chapter cites without re-deriving.

Difficulty

The natural first attempt treats the worst-case risk as an instance of the general finite convex reduction (Theorem 8, which needs $\ell$ or a related concave/convex-conjugate structure and applies for a general norm exponent $p,q$ with $1/p+1/q=1$) and specializes it to $p=1$. This is not the paper's own route for Theorem 10: at $p=1$ the dual exponent $q=\infty$, and the general finite-convex-program reduction (11) degenerates in a way that is more naturally derived directly from the type-1 Wasserstein distance's own dual (Kantorovich-Rubinstein) representation — a Lipschitz test function pairs exactly with a type-1 transportation cost, which is why the answer is Lipschitz-regularization and not some other penalty. The Lean statement records only the final equality (the proof itself, via Theorem 8 or the direct Kantorovich-Rubinstein route, is left as sorry, as required for a draft item), but the choice of which milestones would support that proof is exactly this: the type-1/Lipschitz-modulus duality is a structurally different argument from the finite-convex-program machinery Theorem 8 needs for other $p$, which is why this chapter's earlier draft mistakenly tried to reach a specialization of Theorem 10 (Proposition 2, restated for a labeled, $\kappa=\infty$ ambiguity set) through a finite-atom shortcut that does not actually capture the general type-1 ball Theorem 10's own proof needs — see Formalization scope.

Formalization scope

$\Xi$ is EuclideanSpace ℝ (Fin m) (fixed to Set.univ, matching the theorem's own "Ξ = R^m" hypothesis) with an arbitrary fixed norm as a type-class parameter, not specialized to Euclidean. The worst-case risk, Wasserstein distance, ambiguity set, nominal risk, empirical distribution and Lipschitz modulus are all redefined locally in this chapter's own namespace, verbatim restatements of 01-duality's own drafts of the same objects (identical EReal/ENNReal/Integrable conventions), because 01-duality is not yet a published mission (CAPTAIN_ADDENDUM_WAVE2.md rule 5); a later upload can retire the duplication once it is live. hN : 0 < N excludes the empty-sample case the paper's own "N training samples" presupposes; hℓ : Integrable ℓ (empiricalDistribution ξhat) guards the empirical risk's Bochner integral (automatically satisfied under any real-valued ℓ since the empirical distribution is a finite convex combination of Dirac masses, but stated explicitly rather than assumed silently). No constant is pinned to a numeral. This mission originally targeted Proposition 2 (regularization by robustification for classification, p. 25), a specialization of Theorem 10 to a labeled, κ=∞ input-output ambiguity set — but that draft's κ=∞ ambiguity set restricted admissible perturbations to a finite family of exactly N atoms, each sample's entire mass moving as one indivisible unit, which is a strict, non-faithful subset of the true κ=∞ Wasserstein ball (splitting a sample's mass across several destinations is a legitimate, and for a convex loss strictly more powerful, perturbation by Jensen's inequality). This made the goal theorem false for at least one of the three loss functions (Table 1) the mission itself certified as covered (logloss), not merely narrower than the paper's claim — moderation caught this and required either a faithful general-ambiguity-set reformulation or retargeting the goal. This mission takes the latter: Theorem 10 itself, already correctly and generally stated using the unrestricted Wasserstein ball throughout, is now the goal, with no further milestone (no other numbered result in this chunk's scope is needed for Theorem 10's own statement beyond the shared definitions above). A future mission in this series can revisit Proposition 2/3 with a properly general labeled ambiguity set once that construction (label-pinned marginal plus an aggregate coupling condition on the input coordinate, rather than a finite atom family) is built.

Selected references

• Kuhn, D., Mohajerin Esfahani, P., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. INFORMS TutORials in Operations Research. https://doi.org/10.1287/educ.2019.0198
• Shafieezadeh-Abadeh, S., Esfahani, P. M., & Kuhn, D. (2015). Distributionally robust logistic regression. Advances in Neural Information Processing Systems, 28.
• Blanchet, J., Kang, Y., & Murthy, K. (2019). Robust Wasserstein profile inference and applications to machine learning. Journal of Applied Probability, 56(3), 830–857.

Motivation

Minimum mean square error (MMSE) estimation — predicting a signal $x$ from a noisy observation $y$ by minimizing expected squared prediction error — underlies linear systems theory, linear regression, Kalman filtering, and multiple-input multiple-output signal processing. Its classical solution assumes the joint distribution of $(x,y)$ is known exactly; in practice it is estimated from data, and the estimator inherits sampling error and model risk. Kuhn, Mohajerin Esfahani, Nguyen & Shafieezadeh-Abadeh's 2019 INFORMS TutORials chapter shows that hedging the MMSE objective against every distribution in a Wasserstein ball around the empirical distribution — an infinite-dimensional worst case over an intractable set of measures, a priori — collapses to a tractable, finite-dimensional convex semidefinite program (Theorem 25, p. 29), building on the Gelbrich-hull machinery of Section 2.3. This mission formalizes that reduction.

Setting

Fix $m_x, m_y \in \mathbb{N}$ and let $\xi = (x,y) \in \mathbb{R}^{m_x} \times \mathbb{R}^{m_y}$ be a random vector: $x$ the signal to be estimated, $y$ the observation. An estimator is a measurable function $\psi : \mathbb{R}^{m_y} \to \mathbb{R}^{m_x}$; write $\Psi$ for the family of all estimators. The distribution of $\xi$ is only known to lie in a type-2 Wasserstein ball $B_{\varepsilon,2}(\hat P_N)$ centered at an elliptical nominal distribution $\hat P_N = E_g(\hat\mu,\hat\Sigma)$ with nominal mean $\hat\mu \in \mathbb{R}^m$ ($m=m_x+m_y$), nominal covariance $\hat\Sigma \in S^m_+$, and density generator $g$. The distributionally robust MMSE estimation problem is

$$\inf_{\psi \in \Psi} \sup_{Q \in B_{\varepsilon,2}(\hat P_N)} E_Q\big[\|x-\psi(y)\|_2^2\big]. \tag{35}$$

Writing $\hat\Sigma = \begin{pmatrix}\hat\Sigma_{xx}&\hat\Sigma_{xy}\\\hat\Sigma_{yx}& \hat\Sigma_{yy}\end{pmatrix}$ blockwise, the nonlinear convex SDP

$$\max_S f(S) = \mathrm{Tr}[S_{xx} - S_{xy}S_{yy}^{-1}S_{yx}] \quad \text{s.t.} \quad S = \begin{pmatrix}S_{xx}&S_{xy}\\S_{yx}&S_{yy}\end{pmatrix} \succeq 0,\; S_{xx} \succeq 0,\; S_{yy} \succeq 0,\; \mathrm{Tr}[S+\hat\Sigma-2(\hat\Sigma^{1/2}S\hat\Sigma^{1/2})^{1/2}] \le \varepsilon^2,\; S \succeq \lambda_{\min}(\hat\Sigma) I \tag{36}$$

is the finite-dimensional relaxation the chapter builds toward.

Formalization targets

Goal (Theorem 25, distributionally robust MMSE estimator). If $\hat\Sigma \succ 0$, then the optimal value of problem (35) equals the optimal value of SDP (36). Moreover, if $S^\star$ is optimal in (36) with $S^\star_{yy}$ invertible, then the affine function

$$\psi^\star(y) = S^\star_{xy}(S^\star_{yy})^{-1}(y-\hat\mu_y) + \hat\mu_x$$

attains the outer infimum of (35) — it is a distributionally robust MMSE estimator, exhibited in closed form from an SDP optimizer.

Significance

Theorem 25 reduces an a priori infinite-dimensional, worst-case functional optimization problem (an infimum over all measurable estimators of a supremum over all distributions within a Wasserstein ball) to a finite convex program with one linear matrix inequality, one Loewner-order lower bound, and one trace/matrix-square-root constraint — solvable in polynomial time, with the optimal estimator recovered in closed form from the SDP's optimal block matrix. It shows that robustifying MMSE estimation against distributional ambiguity does not sacrifice tractability: the resulting estimator remains affine, the same functional form as the classical (non-robust) best linear unbiased estimator, only with its coefficients drawn from a regularized covariance estimate rather than the raw sample covariance. Formalizing it fixes, machine-checkably, the exact shape of that regularization — which SDP constraints are load-bearing (the Loewner lower bound in particular rules out a numerically unstable near-singular $S_{yy}$) and which conditions ($\hat\Sigma \succ 0$, $S^\star_{yy}$ invertible) the closed-form estimator formula actually needs.

Difficulty

The paper's own remark (p. 29) names the two nontrivial steps: first, "establishing a minimax theorem for (35) and exploiting the properties of elliptical distributions" to show the outer infimum is attained by an affine estimator — a priori (35) ranges over all measurable $\psi$, and there is no obvious reason the worst case forces linearity. Second, "combining this structural insight with Theorem 16" (the SDP-representability result for indefinite quadratic losses under an elliptical nominal distribution, itself a nontrivial closed-form reduction of an infinite-dimensional worst-case risk) to convert the now-restricted problem over affine estimators into the finite SDP (36). Neither step is a routine consequence of the ambiguity-set definitions alone; each requires structural facts about elliptical distributions and quadratic losses proved earlier in the chapter.

Formalization scope

The signal-observation space is EuclideanSpace ℝ (Fin mx ⊕ Fin my), with $x$ and $y$ recovered as the two summand projections; the block matrix $S$ is Matrix (Fin mx ⊕ Fin my) (Fin mx ⊕ Fin my) ℝ, and Matrix.toBlocks₁₁/toBlocks₁₂/toBlocks₂₁/toBlocks₂₂ give its four blocks. The constraint "$S_{xy}=S_{yx}^\top$" is not stated as a separate hypothesis: it follows automatically once $S$ is symmetric (implied by S.PosSemidef), so encoding the feasible set from a single symmetric S rather than four independently-quantified blocks makes it structurally impossible to drop — see pitfall 4 of BRIEF.md. The outer infimum of problem (35) ranges only over measurable $\psi$ (Measurable ψ on the binder), matching the paper's own definition of $\Psi$ as "the family of all possible measurable estimators" (p. 29) exactly. λ_min(Σ̂) is taken as a hypothesis parameter characterized by the two properties that make it the minimum ("≤ every eigenvalue of Σ̂, and attained by some eigenvalue"), rather than invoking a specific Mathlib min-eigenvalue API by name. S_{yy}⁻¹ uses the ordinary matrix inverse (junk zero matrix when singular), matching the paper's literal notation; S^\star_{yy} invertible is stated as an added hypothesis, not present verbatim on the page, because the paper leaves the formula's well-definedness implicit — disclosed per pitfall 5 rather than silently assumed away. The paper's own "which is always solvable" clause is not asserted: Theorem 25 states, as part of itself, that SDP (36) attains its maximum (an unconditional existence claim for an optimal $S^\star$); this formalization states only the conditional consequences of such an $S^\star$ existing, not that one does — proving or asserting solvability is out of this mission's scope, so the Lean statement is strictly weaker than Theorem 25's own conclusion on this point, disclosed rather than silently dropped. All risk-style suprema are EReal-valued and Integrable-guarded, matching the series' convention. No milestone theorem is included: the paper's own proof sketch derives (33)'s and by extension (36)'s SDP "via Theorem 16", but Theorem 16 (indefinite quadratic loss and $p=2$, eq. 23) was itself judged too heavy to state faithfully in 02-gelbrich's time budget and is not redefined here either — see STATUS.md. Theorem 24 (the Wasserstein shrinkage estimator, this chapter's originally recommended goal) is out of scope: its closed-form eigenvalue transformation (eq. 34a/34b) requires transcribing nested square roots from a rendered PDF page that this session's time budget did not allow verifying to the standard the brief demands (pitfall 1); the brief's own documented fallback to Theorem 25 was taken instead.

Selected references

• Kuhn, D., Mohajerin Esfahani, P., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. INFORMS TutORials in Operations Research. https://doi.org/10.1287/educ.2019.0198
• Nguyen, V. A., Shafieezadeh-Abadeh, S., Yue, M.-C., Kuhn, D., & Wiesemann, W. (2021). Optimistic distributionally robust optimization for nonparametric likelihood approximation. Advances in Neural Information Processing Systems, 32.
• Shafieezadeh-Abadeh, S., Nguyen, V. A., Kuhn, D., & Mohajerin Esfahani, P. (2018). Wasserstein distributionally robust Kalman filtering. Advances in Neural Information Processing Systems, 31.

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]).

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).

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

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

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).

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).

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

Motivation

When several processors share a workload, the first question is how long the workload takes if it is spread out as well as possible. If a job may be interrupted and resumed later, possibly on another processor — preemption, the setting of operating systems, of communication links and of any resource that can be time-shared — the answer is a closed formula, and it is one of the oldest results in scheduling: McNaughton's wrap-around rule of 1959 (Scheduling with deadlines and loss functions, Management Science 6, doi:10.1287/mnsc.6.1.1) shows that on identical machines the optimal makespan is the larger of the longest job and the average load. Horvath, Lam and Sethi (A level algorithm for preemptive scheduling, Journal of the ACM 24, 1977, doi:10.1145/321992.321995) extended this to machines of different speeds, and Gonzalez and Sahni (Preemptive scheduling of uniform processor systems, Journal of the ACM 25, 1978, doi:10.1145/322047.322055) gave the fast algorithm with few preemptions. Brucker's Chapter 5 (doi:10.1007/978-3-540-69516-5) presents the level-algorithm version, and this mission formalizes the statements it proves about schedules.

Setting

There are $n$ jobs with processing requirements $p_1,\dots,p_n>0$ and $m$ uniform machines with speeds $s_1,\dots,s_m>0$: running job $i$ on machine $j$ for a period of length $\ell$ performs $s_j\ell$ units of its requirement, so the whole job would take $p_i/s_j$ time units there. Identical machines are the case $s_1=\dots=s_m=1$.

A preemptive schedule is a finite list of pieces, each a job, a machine, a start time and a stop time. It is feasible for the data $(s,p)$ when every piece lies in $[0,\infty)$, no two pieces on the same machine overlap, no two pieces of the same job overlap (a job is on at most one machine at any instant), and every job $i$ receives total work exactly $p_i$ over its pieces. Its makespan $C_{\max}$ is the largest stop time; the completion time $C_i$ of job $i$ is the largest stop time of one of its pieces. A schedule is nonpreemptive when every job consists of a single piece.

Following Section 5.1.2 the data are sorted, $p_1\ge\dots\ge p_n$ and $s_1\ge\dots\ge s_m$, with $n\ge m$, and one writes $P_j=\sum_{i\le j}p_i$, $S_j=\sum_{i\le j}s_i$. For a set $A$ of jobs, $h(A)=S_{|A|}$ if $|A|\le m$ and $h(A)=S_m$ otherwise: the largest combined speed that $|A|$ jobs can use at one instant.

Formalization targets

Goal — Theorem 5.8 (printed p. 127)

The optimal makespan of $Q\mid pmtn\mid C_{\max}$ is the bound (5.5):

$$w \;=\; \max\Bigl\{\max_{j=1}^{m-1}\frac{P_j}{S_j},\ \frac{P_n}{S_m}\Bigr\} ,$$

in the sense that some feasible preemptive schedule has makespan exactly $w$ and no feasible preemptive schedule has a smaller makespan.

The lower bound (5.5) (printed p. 125)

Every feasible preemptive schedule has makespan at least $w$.

$P\mid pmtn\mid C_{\max}$ (printed p. 108)

On identical machines, $LB=\max\{\max_i p_i,\ \tfrac1m\sum_i p_i\}$ is a lower bound on the makespan and is attained by some feasible preemptive schedule.

Condition (5.8) (printed p. 129)

The jobs can be scheduled preemptively within $[0,T]$ if and only if $\sum_{i\in A}p_i\le T\,h(A)$ for every set $A$ of jobs.

Theorem 5.7 (printed p. 121)

For $P\mid pmtn\mid\sum w_iC_i$ with nonnegative weights there is an optimal schedule without preemption.

Significance

Theorem 5.8 turns an optimization over an infinite family of schedules into a formula in the data, and the formula is tight in both directions: each of its terms is a resource bound that some schedule meets exactly. That is what makes preemptive makespan minimization one of the few parallel-machine problems that is solvable at all — its nonpreemptive counterpart $P2\parallel C_{\max}$ is NP-hard (p. 124) — and it is why the preemptive relaxation appears as a bound inside branch-and-bound methods for the nonpreemptive problems.

Condition (5.8) is the form in which the result is reused. It is a Hall-type condition, one inequality per set of jobs, and it is exactly what Section 5.1.2 needs to prove Theorem 5.9, which decides $Q\mid pmtn; r_i\mid L_{\max}$ by a maximum flow in an expanded network. Theorem 5.7 is the complementary statement for the other classical objective: for total weighted completion time preemption buys nothing, so the nonpreemptive solutions of Section 5.1.1 are optimal in the larger class too.

On status: every statement here is classical and proved, and the formalization adds a checked model of preemptive schedules. Mathlib has no scheduling material, and the platform's SchedulingAlgorithms series so far models only single-machine sequences (missions I, II, IV) and two-machine permutation flow shops (mission III), none of which allow a job to be split. The piece-list model of this mission is the first reusable object for preemptive and parallel-machine problems, and the later sections of Chapter 5 — $Q\mid pmtn; r_i\mid L_{\max}$, $P\mid pmtn\mid L_{\max}$ — are stated in it.

Difficulty

The obvious first idea for the goal is to run McNaughton's rule with the speeds ignored. It fails on uniform machines: filling machines one after another does not respect the constraint that a long job on a slow machine is not done when a short job on a fast one is. The correct idea is the level algorithm — always process the jobs of highest remaining requirement on the fastest free machines, sharing machines among tied jobs — and the difficulty is in the analysis rather than the idea. The proof of Theorem 5.8 has to show that the schedule it produces ends exactly at one of the terms of $w$: either no machine idles before the end, giving $P_n/S_m$, or the machines finish in speed order with the first $j$ jobs busy from time $0$, giving $P_j/S_j$. Making that case analysis rigorous requires tracking that the order of remaining requirements is preserved over time (the invariant (5.6)) and that ties are broken consistently.

A second, formal difficulty is that the level algorithm's output is defined by continuous-time events (the next completion, the next time two levels coincide), so producing an explicit finite list of pieces with the required properties is itself a construction. Any proof must build a concrete schedule; "the infimum of makespans equals $w$" is not the goal.

For the lower bound the trap is the opposite: it is tempting to argue only with total capacity $S_mT$, which gives $P_n/S_m$ but not $P_j/S_j$. The latter needs the rule that a job is on at most one machine at a time, so that $j$ jobs run at combined speed at most $S_j$; a model that let a job be split across machines simultaneously would make the theorem false, and the definition of feasibility rules it out explicitly.

Formalization scope

A schedule is a List of Pieces over jobs Fin n and machines Fin m, with real start and stop times. Feasibility is the three-part condition of the Setting, disjointness of two pieces meaning one stops no later than the other starts. Work is measured with the machine's speed, so the same definitions cover identical machines as the constant speed $1$. Pieces of length zero and unsorted lists are allowed; both are harmless.

The sorted orders are hypotheses Antitone p and Antitone s, the speeds and requirements are positive, $m\ge 1$ and, where the book assumes it, $n\ge m$. The book's normalization $s_1=1$ is not assumed: every statement here is invariant under scaling all speeds, and the book uses the normalization only for a running-time estimate. The bound $w$ is defined as the maximum of an explicit nonempty finite set, so no supremum of an empty or unbounded set occurs; $LB$ takes a proof that $n\ge 1$ so that $\max_i p_i$ is meaningful.

Two things are deliberately not stated. The level algorithm itself is not transcribed: Theorem 5.8 is stated as the existence of an optimal schedule of makespan $w$, which is what its proof establishes. And Theorem 5.9, the flow characterization for $Q\mid pmtn; r_i\mid L_{\max}$, is left for a later mission, since it needs release times and the expanded network on top of this model.

A trivializing reading is excluded by the existential form of the goal and of Theorem 5.7: each asserts that an optimal schedule exists, not merely that any optimal schedule has a property. A proof of the goal has to construct a schedule; a proof of Theorem 5.7 has to construct a nonpreemptive one that beats every preemptive competitor. Contributions welcome beyond the milestones: a general lemma that a feasible schedule can be normalized to sorted, positive-length pieces, and a proof that (5.8) for the sets $\{1,\dots,j\}$ is equivalent to $w\le T$.

Selected references

• Peter Brucker, Scheduling Algorithms, 5th ed., Springer, 2007, Chapter 5. doi:10.1007/978-3-540-69516-5
• Robert McNaughton, Scheduling with deadlines and loss functions, Management Science 6 (1959). doi:10.1287/mnsc.6.1.1
• E. C. Horvath, S. Lam and R. Sethi, A level algorithm for preemptive scheduling, Journal of the ACM 24 (1977). doi:10.1145/321992.321995
• Teofilo Gonzalez and Sartaj Sahni, Preemptive scheduling of uniform processor systems, Journal of the ACM 25 (1978). doi:10.1145/322047.322055