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arxiv: 2604.25764 · v1 · submitted 2026-04-28 · 🧮 math.OC

Benders Cut Filtering for Affine Potential-Based Flow Problems with Robustness Scenarios and Topology Switching

Tim Donkiewicz , Oliver Gaul This is my paper

Pith reviewed 2026-05-07 15:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords Benders decompositioncut filteringaffine potential-based flowrobustness scenariostopology switchingmixed-integer linear programmingoptimization algorithms
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The pith

Benders cut filtering via violation and k-medoids clustering cuts solve time by 57 percent on affine flow problems.

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

The paper studies which Benders cuts to add to the master problem when many scenario subproblems generate cuts in one iteration. It proposes three informed filtering approaches: keeping the most violated cuts, selecting diverse cuts through k-medoids clustering on cosine distances, and a hybrid that picks one most-violated cut per cluster. Each can be paired with an aggregated cut to retain information from the discarded ones. Experiments on 149 affine potential-based flow instances with topology switching and robustness scenarios show that all three strategies solve substantially more instances than the unfiltered baseline while reducing geometric mean solve time by 55 to 57 percent.

Core claim

For Benders decomposition applied to affine potential-based flow problems with robustness scenarios and topology switching, filtering the cuts generated in each iteration by retaining the most-violated cuts, by selecting diverse cuts through k-medoids clustering on pairwise cosine distances, or by a hybrid of the two approaches, optionally augmented with an aggregated cut, allows the algorithm to solve at least 125 of 149 test instances compared with 91 for the unfiltered baseline and reduces the shifted geometric mean solve time from 629.34 s to 271.89 s under the hybrid strategy.

What carries the argument

Violation-based, diversity-based (k-medoids on cosine distance), and hybrid Benders cut filtering strategies, optionally augmented with an aggregated cut that preserves information from discarded cuts.

If this is right

  • Adding fewer cuts per iteration reduces the total number of constraints in the master problem even if the iteration count rises slightly.
  • Informed filtering strategies solve at least 34 more instances than adding every generated cut.
  • The hybrid strategy that combines violation and diversity criteria produces the largest reduction in geometric mean solve time.
  • An aggregated cut can be added alongside filtered cuts to retain information without increasing master-problem size proportionally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same filtering logic could be applied to Benders decompositions arising in other network design or robust optimization settings.
  • Embedding these filters directly inside commercial MIP solvers might automate cut management for users who employ decomposition.
  • Alternative clustering algorithms or distance measures on the cut space might yield further reductions in solve time.
  • The approach may interact productively with other acceleration techniques such as cut strengthening or warm-starting of the master problem.

Load-bearing premise

The 149 instances of the affine potential-based flow problem with topology switching and robustness scenarios are representative enough that the observed speedups will generalize to other problem classes or larger real-world instances.

What would settle it

Running the same filtering strategies on a different family of Benders-decomposable problems or on substantially larger instances and finding that the number of solved instances or the geometric mean solve time shows no improvement or becomes worse than the unfiltered baseline.

Figures

Figures reproduced from arXiv: 2604.25764 by Oliver Gaul, Tim Donkiewicz.

Figure 1
Figure 1. Figure 1: Distribution of solve times across Benders cut filtering strategies on view at source ↗
Figure 2
Figure 2. Figure 2: Performance profile of solve times across Benders cut filtering strategies ( view at source ↗
Figure 3
Figure 3. Figure 3: Parameter search on 20 validation instances. Each cell shows the ratio of shifted view at source ↗
read the original abstract

Many large-scale optimization problems decompose into a master problem and scenario subproblems, a structure that can be exploited by Benders decomposition. In Benders decomposition, each iteration may generate many cuts from scenario subproblems, and adding all of them as constraints then causes the master problem to grow rapidly. These are constraints that may need to be added to the master problem to guarantee optimality and feasibility of solutions, but we can avoid adding those constraints that are never violated. Adding fewer cuts per iteration can reduce the number of cuts added in total, but increase the number of iterations. In contrast, the cuts filtered for regular cut selection in mixed-integer programming solvers are optional and added exclusively to improve runtime behavior. We study Benders cut filtering: given the Benders cuts produced in an iteration, which subset should be added to the master problem? To our knowledge, few prior works have studied this question. We propose violation-based filtering (retaining the most-violated cuts), diversity-based filtering via k-medoids clustering on pairwise cosine distances (adding an original cut closest to the cluster centroid), and a hybrid that selects a most-violated cut per cluster. Each strategy can be augmented with an aggregated cut that retains discarded information. Computational experiments on 149 instances of an affine potential-based flow problem with topology switching and robustness scenarios -- solved via Benders decomposition -- show that all informed filtering strategies solve at least 125 instances (vs. 91 for the unfiltered baseline), reducing shifted geometric mean solve time by 55-57%. The hybrid strategy attains the best geometric mean (271.89 s vs. 629.34 s, a 57% reduction, p < 0.001).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes violation-based, diversity-based (k-medoids clustering on pairwise cosine distances), and hybrid Benders cut filtering strategies for affine potential-based flow problems with robustness scenarios and topology switching. Each strategy can be augmented with an aggregated cut. Computational experiments on 149 instances show that all informed filters solve at least 125 instances (vs. 91 for the unfiltered baseline), with the hybrid yielding the lowest shifted geometric mean time of 271.89 s (57% reduction, p < 0.001).

Significance. If the empirical results hold, the work provides concrete, statistically supported evidence that informed cut filtering can substantially improve Benders decomposition performance on this class of network flow problems by limiting master-problem growth while preserving solution quality. The use of shifted geometric means, solved-instance counts, and p-value testing on a fixed test set strengthens the practical contribution for similar decomposable problems.

minor comments (2)
  1. [Abstract] Abstract: the statement that 'few prior works have studied this question' would benefit from 1-2 citations to existing Benders cut management or cut selection literature to better situate the novelty.
  2. [Computational Experiments] Computational experiments section: more detail on how the 149 instances were generated or selected would help readers assess the scope of the observed speedups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on Benders cut filtering strategies and for recognizing its practical significance in improving decomposition performance on affine potential-based flow problems. We appreciate the recommendation for minor revision and will incorporate any editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No circularity: direct empirical comparison on fixed instances

full rationale

The paper proposes three Benders cut filtering strategies (violation-based, k-medoids diversity, hybrid) plus optional aggregation, then reports concrete runtime and solvability results on a fixed collection of 149 instances. These outcomes are obtained by direct execution of the algorithms inside a standard Benders loop; they do not rely on any fitted parameter renamed as a prediction, any self-definitional equation, or any load-bearing self-citation chain. The abstract and skeptic analysis confirm that the reported geometric-mean speedups and instance counts follow immediately from applying the described filters to the given test set, with no hidden derivation that reduces to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The approach assumes standard Benders decomposition validity for the affine potential-based flow problem with robustness scenarios and topology switching; no new free parameters, axioms, or invented entities are introduced beyond the filtering heuristics themselves.

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