ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization

Alan Kuhnle; Joan Vendrell Gallart; Solmaz Kia

REVIEW 4 major objections 5 minor 16 references

ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read ResQue Greedy claims a local curvature trigger can steer sequential greedy to better solutions while keeping its worst-case guarantee.

desk verdict A practical greedy tweak that keeps the standard worst-case bound, but the trigger that is supposed to make it better is unproven and the experiments lack statistical support. read the letter →

arxiv 2505.13670 v1 pith:ERSS7W2M submitted 2025-05-19 cs.DM cs.DSmath.OC

Joan Vendrell Gallart , Alan Kuhnle , Solmaz Kia This is my paper · ORCID

classification cs.DMcs.DSmath.OC MSC 68W2590C27

keywords submodularmaximizationsequentialgreedyalgorithmcurvaturecardinalityconstraintrewiringsetfunctioncoveragelattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that a small, curvature-guided modification to the classic sequential greedy algorithm for maximizing a monotone submodular function under a cardinality constraint can produce higher-value solutions than ordinary greedy, while keeping the same worst-case approximation guarantee. The proposed method, Rewired Sequential Greedy (ResQue Greedy), treats the greedy construction as a path through the lattice of subsets, watches a local set-curvature statistic, and when an 'anomaly' is detected it removes the past element that most degraded future marginal gains and restarts the next greedy step from the reduced set. Theorem 4.1 preserves the standard $(1 - 1/e)$ guarantee for any choice of trigger and removal rule, and the reported experiments show higher coverage and feature counts with a modest increase in oracle queries. If the method works as claimed, it is an inexpensive drop-in improvement over one of the most widely used algorithms for resource allocation problems.

What carries the argument

The load-bearing object is the set curvature $$\gamma(S \mid A) = 1 - \min_{e \in A} \frac{f(S \cup \{e\}) - f(S)}{f(\{e\}) - f(\emptyset)},$$ a local version of total curvature that quantifies how much adding elements from $A$ to the already-chosen set $S$ is worth relative to adding them to the empty set. Along a sequential construction it generates two running statistics: expansion curvature $\gamma_e(S_i)$, the maximum set curvature encountered so far, and path curvature $\gamma_p(S_i)$, the maximum curvature of the individual chosen elements. The mechanism that carries the argument is the pair of Definitions 4 and 5: the trigger law fires when $\gamma(S_i \mid A_i) \le \gamma_e(S_i)$, declaring that an earlier element has harmed the current expansion, and the step-back policy removes the element corresponding to the largest historical set curvature before running one more greedy iteration. Curvature is used as a cheap, query-free predictor of which past choices are worth undoing.

What would settle it

One concrete test: on every monotone submodular instance over a small ground set (for example $|P| \le 7$, enumerated exhaustively), run the ordinary greedy algorithm and ResQue Greedy and record every trigger firing; if any instance has a firing after which the final ResQue Greedy value is strictly below the ordinary greedy value, or a firing where deleting the element identified by the step-back policy and re-greedying yields no value gain at that stage, the claimed work of the trigger is not happening.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that local curvature can be used online to correct the shortsightedness of sequential greedy. For a set $S$ and a permissible expansion $A$, set curvature $\gamma(S \mid A)$ measures how much marginal gains of elements in $A$ have already been diminished by the choice of $S$. Along the greedy path the paper tracks the expansion curvature $\gamma_e(S_i)$, the maximum set curvature seen so far, and declares an anomaly at stage $i+1$ when $\gamma(S_i \mid A_i) \le \gamma_e(S_i)$, because a past element appears responsible for the erosion of future gains. The step-back policy then discards the element that produced that historical maximum and re-runs one greedy selection from the smaller set. Theorem 4.1 proves that Algorithm 2 retains the worst-case bound $f(S_{\mathrm{RSG}}) \ge (1 - 1/e)f(S^*)$ regardless of the trigger or step-back rule, and Theorem 4.2 gives an if-and-only-if path-curvature condition under which the rewired path stays at least as good as the ordinary greedy path. The two experiments report average normalized coverage rising from $0.7823$ for greedy to $0.8874$ for ResQue Greedy, and Mars features found rising from $6{,}658$ to $7{,}714$, while a random-rewiring control scores $0.7439$ and $6{,}853$, respectively.

Load-bearing premise

The load-bearing premise is that the curvature trigger identifies past greedy choices whose removal improves the eventual solution; the paper motivates this with intuition but does not prove it, and the random-rewiring baseline shows that an uninformative trigger can make rewiring worse than plain greedy.

Editorial extensions

If this is right

Any instance that can be fed to sequential greedy can instead be passed to ResQue Greedy without losing the $(1 - 1/e)$ worst-case guarantee, since Theorem 4.1 holds for arbitrary later-stage trigger and step-back rules.
When the trigger fires rarely, the extra cost is one additional greedy pass per rewiring, so the method remains polynomial-time; in the two experiments it used roughly 14–18 percent more oracle queries than plain greedy.
Ordinary greedy is not a lower bound on the rewired path in general: the paper's random-rewiring control (0.7439 normalized coverage) lands below plain greedy (0.7823), so the curvature trigger, not the act of rewiring, is what must do the work.
For functions with low total curvature, the paper expects little gain because the trigger rarely fires; in such cases ResQue Greedy behaves essentially like sequential greedy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One implication the authors leave implicit: the trigger compares curvatures but never checks whether the suspected element's actual marginal contribution at the current stage is small, so a variant that fires only when the gap $\gamma_e(S_i) - \gamma(S_i \mid A_i)$ exceeds a tuned margin might buy larger gains per extra query.
Because a rewiring at the final step re-selects a best element from the reduced set, it cannot lower the value at that step; a cheaper variant that skips curvature monitoring and always performs one final re-greedy pass is a natural baseline the paper does not isolate.
The midpoint update for curvatures after a step-back, equation (12), is an approximation, so over multiple rewires the trigger's statistics drift from the true ones; tracking exact curvatures after each rewire would test whether the heuristic's performance degrades on long horizons.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: ResQue Greedy is a simple, cheap modification of the standard greedy algorithm for submodular maximization under a cardinality constraint. The authors add a curvature-based trigger and a step-back/rewiring step, prove that the rewired version still has the classic (1-1/e) worst-case bound, and report noticeably higher objective values on two coverage problems. The catch is that the trigger is a heuristic with no supporting analysis, and there is a concrete reason to expect it to fire on benign events.

The algorithm itself is new as far as I know. I have not seen a step-back policy driven by set and expansion curvature in the submodular maximization literature, and the lattice/path framing is a genuinely different way to present greedy. Theorem 4.1 is correct: after a step-back, the new greedy choice has marginal gain at least that of the removed element, so the value does not drop, and the induction goes through. The worst-case guarantee is safe regardless of the trigger. The random-rewiring control in Table I is also a nice experimental idea, because it shows that naive rewiring can hurt.

The soft spots are real. The trigger law in Definition 4 is justified only by intuition. The stress-test concern about set curvature is valid: gamma(S|A) is monotone in S for fixed A, but A shrinks as greedy proceeds. The min over a smaller permissible set can go up, so gamma(S_i|A_i) <= gamma_e(S_i) can occur just because the element achieving the historical minimum was selected, not because an earlier choice was harmful. That is a plausible source of false positives. Theorem 4.2 does not fix this: the notation gamma_p(S^{SG}_{m-1} | s^{SG}_m) is not defined in Definition 3, and the claimed if-and-only-if condition is not shown to be satisfied by the trigger. It reads more like a restatement than a proof of superiority.

There is also a clear error in the complexity discussion: each rewiring costs O(n) additional queries, and with at most k possible triggers the total extra cost is O(kn), not exponential. That should be corrected. The experiments are suggestive but thin: no error bars, no significance tests, and no comparison to standard local search or lazy-greedy baselines. The reported gains are large, so I would not dismiss them, but I also cannot tell from the paper whether they are robust.

Who is this for? Practitioners who want a cheap drop-in improvement over greedy and can tolerate an unproven trigger. Theorists will not find a new bound, because the only proven guarantee is the standard one.

My recommendation: send it to peer review, but require a fix of the complexity statement, a clear statement that the improvement is heuristic-only, and either a better-justified trigger or experiments with variance and a stronger baseline. It is not ready as is, but it deserves a serious referee.

Referee Report

4 major / 5 minor

Summary. The paper proposes ResQue Greedy, a modification of the standard sequential greedy algorithm for monotone submodular maximization under a cardinality constraint. The algorithm adds a rewiring step: when a curvature-based trigger condition (Definition 4) fires, it removes one previously selected element according to a step-back policy (Definition 5) and performs an additional greedy selection. The paper claims that this procedure preserves the standard (1 - 1/e) worst-case approximation guarantee while empirically obtaining higher objective values than standard greedy on two coverage problems. Theorem 4.1 proves the worst-case bound for any trigger and step-back rule, and the argument in that proof is correct. The claimed improvement over greedy, however, rests on the heuristic trigger law and is not established theoretically; Theorem 4.2 is the only theoretical statement about improvement, and it has notational and logical problems. The experimental section provides two coverage examples with no error bars, significance tests, or comparison to standard submodular maximization baselines.

Significance. If the empirical improvement is real, the paper offers a simple, drop-in heuristic that preserves the standard worst-case guarantee and may improve solution quality in practice at modest computational cost. The idea of using local curvature as a trigger for rewiring is interesting and the preservation of the (1 - 1/e) bound for arbitrary trigger and removal rules is a useful robustness property. The theoretical contribution is otherwise modest: Theorem 4.1 reproduces the standard greedy bound, and no result in the paper proves that the proposed trigger law improves the final solution. The paper would be strengthened by releasing code and data, by reporting variance and significance statistics, and by clearly positioning the improvement claim as empirical rather than as a proven tighter bound.

major comments (4)

[Section IV-B, Theorem 4.2] Theorem 4.2 uses the undefined quantity γ_p(S^SG_{m-1} | s^SG_m). Definition 3 defines path curvature γ_p(S_i) only for a whole sequential construction, not for a set-element pair. In addition, the derived condition involves γ(S^RSG_{m-1} | A^SG_{m-1}), whereas the preceding calculation requires γ(S^RSG_{m-1} | {s^SG_m}). Since γ(S|A) ≥ γ(S|{e}) for any e in A, the displayed condition is merely sufficient, not necessary. The 'if and only if' claim is therefore not justified, and the proof also silently assumes that s^SG_m belongs to P \ S^RSG_{m-1}, which may fail after a rewiring. This theorem needs to be restated with correct notation and a correct logical status.
[Section IV-A, Definition 4] The trigger law is the load-bearing component of the claimed improvement, but the paper gives no proof that the condition γ(S_i|A_i) ≤ γ_e(S_i) identifies elements whose removal raises the final objective. The motivating argument in Section IV-A assumes that γ(S_i|A_i) grows as elements are added, yet Lemma 3.1 is stated only for a fixed permissible set A; because A_i shrinks at each step, γ(S_i|A_i) can decrease simply because the element attaining the minimum marginal ratio has already been selected. The random-rewiring control in Table I shows that rewiring without an informative trigger can degrade performance, so the trigger law is not a cosmetic detail. The authors should either prove a property of the trigger law that relates it to the condition in Theorem 4.2 or substantially strengthen the empirical case.
[Section V, Tables I and II] The experimental evidence does not statistically support the central claim. Table I reports averages over 100 Monte Carlo runs without error bars, confidence intervals, or significance tests, and Table II is a single instance. There is no comparison to standard baselines such as lazy greedy, local search, or a simple restart heuristic. Moreover, in Table II the random-rewiring greedy achieves 6,853 coverage versus 6,658 for standard greedy, which undermines the narrative that uninformed rewiring necessarily hurts performance. The improvement of ResQue Greedy over greedy needs to be demonstrated with variance statistics and, preferably, with multiple problem classes.
[Abstract and Section I] The abstract and introduction claim that ResQue Greedy 'achieves tighter near-optimality bounds' than sequential greedy. As stated, this is not established: Theorem 4.1 gives exactly the standard (1 - 1/e) bound, and no theorem proves that the algorithm's solution value dominates that of standard greedy. The only support for the tightness claim is empirical. The paper should either prove a formal improvement condition that is actually satisfied by the trigger law, or explicitly reword the contribution as an empirically effective heuristic with a preserved worst-case guarantee.

minor comments (5)

[Section III, Definition 2] The paper uses the term 'normal' for f but never states the normalization condition; it should explicitly say f(∅)=0.
[Lemma 3.2 proof] There is a typo in the proof of part (b): 'γ(Sℓ|{ℓℓ+1})' should be 'γ(Sℓ|{s_{ℓ+1}})'.
[Section IV-B, Eq. (12)] The update rule in Eq. (12) uses A'_ℓ without defining it; please define the permissible set after a step-back.
[Section V, Table II] The 'Queries' column in Table II appears to report total value-query counts, but this is not stated; please clarify how queries are counted and consider reporting standard deviations for the Monte Carlo results.
[Conclusion] The conclusion states that numerical experiments 'confirmed' the effectiveness of the method; given the lack of statistical tests, a softer formulation such as 'suggest' would be more appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rewiring heuristic is evaluated empirically, and the theoretical results are standard or conditional rather than self-referential.

full rationale

The central claim — that ResQue Greedy improves on sequential greedy — rests on an algorithmic heuristic (Definitions 4 and 5) whose trigger and step-back are defined directly from the value oracle f and the resulting set/expansion/path curvatures. No parameter is fitted to the experimental outcomes, and the Monte-Carlo coverage gains in Tables I and II are direct evaluations of the objective, not inversions of a fitted constant. Theorem 4.1 is the standard (1-1/e) induction and holds for any rewiring because greedy re-expansion from S'_i guarantees f(S_{i+1}) >= f(bar S_{i+1}); this proof does not assume the conclusion. Theorem 4.2 is an algebraic comparison of the two algorithms' marginal decompositions; it is conditional and contains undefined notation (gamma_p(S^{SG}_{m-1} | s^{SG}_m) is not defined in Definition 3), so it is a correctness risk rather than a circular step: it does not smuggle the desired inequality in as an input, and the paper does not claim this theorem alone proves the experimental improvement. Self-citations [12] and [13] are used for background and for the standard inequality (1-1/kappa)^kappa <= e^{-1}; they are not load-bearing for the paper's central claim. The heuristic trigger law is motivated by the lower bound f(S_i) >= (1-gamma_p(S_i)) sum f(s), but the paper does not present that bound as a proof that the trigger must improve every instance; the experiments provide external, oracle-based evidence. Therefore the derivation chain is self-contained against the stated value-oracle model, and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The algorithm defines new curvature quantities (set, expansion, and path curvature) but these are functions of f, not new entities requiring independent evidence. There are no fitted numerical parameters: the trigger threshold and the mean-value update are heuristic design choices. The worst-case guarantee (Theorem 4.1) does not depend on the trigger law; the claimed practical improvement does, which is why the trigger and step-back heuristics are recorded as ad hoc axioms.

assumptions (3)

domain assumption f is monotone, normal, submodular, and accessed via a value oracle.
Standard model for submodular maximization, stated in Section I and used throughout.
ad hoc to paper The trigger law (Definition 4) identifies harmful elements: if γ(S_i|A_i) ≤ γ_e(S_i), then an earlier element caused an anomaly and should be removed.
This heuristic underpins the claimed practical improvement. It is motivated by intuition in Section IV-A but never proven; random rewiring in Table I shows rewiring can hurt without a good selection rule.
ad hoc to paper The step-back policy (Definition 5) and the mean-value update (Eq. 12) preserve enough curvature information for subsequent triggers.
After removing an element, the paper does not recompute all set curvatures but approximates them by the midpoint of an interval (Eq. 12), with no error bound. The algorithm's future trigger decisions rely on these approximate values.

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Cite this review

Pith. "Pith review of ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization." pith.science (2026). https://pith.science/paper/ERSS7W2M

@misc{pith2026250513670,
  author       = {Pith},
  title        = {Pith review of: ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERSS7W2M}},
  note         = {Machine review of arXiv:2505.13670}
}

read the original abstract

This paper introduces Rewired Sequential Greedy (ResQue Greedy), an enhanced approach for submodular maximization under cardinality constraints. By integrating a novel set curvature metric within a lattice-based framework, ResQue Greedy identifies and corrects suboptimal decisions made by the standard sequential greedy algorithm. Specifically, a curvature-aware rewiring strategy is employed to dynamically redirect the solution path, leading to improved approximation performance over the conventional sequential greedy algorithm without significantly increasing computational complexity. Numerical experiments demonstrate that ResQue Greedy achieves tighter near-optimality bounds compared to the traditional sequential greedy method.

Figures

Figures reproduced from arXiv: 2505.13670 by the authors.

**Figure 2.** A multi-sensor deployment problem: the possible allocation points are shown by bigger black dots, while the information points are shown by blue dots. In this simple case, we can see that ResQue Greedy’s rewiring at the last step by dropping the second deployment location chosen by the sequential greedy algorithm and retrying to choose another element leads to an improvement: the sequential greedy solution obtains a… view at source ↗

**Figure 3.** Multi-sensor deployment: 100 Monte-Carlo simulation. TABLE II. Results for the second problem. Coverage Time [s] Queries Greedy 6, 658 0.0974 721 ResQue Greedy 7, 714 0.1082 825 Random Rewiring Greedy 6, 853 0.1002 762 Multi-agent monitoring for space exploration: As a second example, we consider a real-world problem in an space exploration setting. In such missions, it is crucial to use resources efficiently to max… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Feature points (blue dots) and potential deployment points (black ⋆). Agent deployments: standard sequential greedy (red), ResQue Greedy allocations (orange), and random rewiring greedy allocations (green). All sensors are homogenous with the same circular range. To vi…

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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R. Li, N. Mehr, and R. Horowitz, “Submodularity of optimal sensor placement for traffic networks,”Transportation Research Part B: Methodological, vol. 171, p. 29–43, May 2023. Fig. 4:Feature points (blue dots) and potential deployment points (black⋆). Agent deployments: standard sequential greedy (red),ResQue Greedyallocations (orange), and random rewirin...

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P. G. Sessa, I. Bogunovic, A. Krause, and M. Kamgarpour, “Online submodular resource allocation with applications to rebalancing shared mobility systems,” inProceedings of the 38th International Conference on Machine Learning(M. Meila and T. Zhang, eds.), vol. 139 ofProceedings of Machine Learning Research, pp. 9455–9464, PMLR, 18–24 Jul 2021

work page 2021

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S. Liu,A Review for Submodular Optimization on Machine Scheduling Problems, p. 252–267. Springer International Publishing, 2020

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G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions—i,” Mathematical Programming, vol. 14, no. 1, pp. 265–294, 1978

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M. Conforti and G. Cornu ´ejols, “Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem,”Discrete applied mathematics, vol. 7, no. 3, pp. 251–274, 1984

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J. V ondr ´ak, “Submodularity and curvature: The optimal algorithm (combinatorial optimization and discrete algorithms),”Research Institute for Mathematical Sciences Kyoto University, vol. 23, pp. 253–266, 2010

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On bounds for greedy schemes in string optimization based on greedy curvatures,

B. Li, B. Van Over, E. K. P. Chong, and A. Pezeshki, “On bounds for greedy schemes in string optimization based on greedy curvatures,” inIEEE Conference on Decision and Control, (Milan, Italy), 2024

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