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arxiv: 1907.06786 · v1 · pith:Y6PCXQRLnew · submitted 2019-07-15 · 💻 cs.DS · math.OC

Some Black-box Reductions for Objective-robust Discrete Optimization Problems Based on their LP-Relaxations

Khaled Elbassioni This is my paper

Pith reviewed 2026-05-24 20:53 UTC · model grok-4.3

classification 💻 cs.DS math.OC
keywords robust discrete optimizationobjective uncertaintylinear programming relaxationintegrality gap verifierblack-box approximationapproximation algorithmsconvex uncertainty set
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The pith

An integrality gap verifier for the non-robust LP relaxation derives approximation algorithms for the objective-robust version.

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

This paper establishes a black-box method to obtain approximation algorithms for robust discrete minimization problems, where the objective coefficients are uncertain over a convex set. It does so by leveraging an integrality gap verifier that exists for the linear programming relaxation of the corresponding non-robust problem. A reader might care because this transfers known approximation results from standard settings to their robust counterparts without redesigning algorithms from scratch. The approach focuses on objective uncertainty rather than other forms like constraint uncertainty.

Core claim

The author claims that given an integrality gap verifier for the LP relaxation of a non-robust discrete minimization problem, one can derive an approximation algorithm for the version where the objective is chosen adversarially from a convex uncertainty set. This reduction works by composing the verifier in a way that accounts for the worst-case objective within the uncertainty set.

What carries the argument

The integrality gap verifier for the LP relaxation, adapted through a black-box reduction to produce a guarantee for the robust objective.

If this is right

  • Many existing approximation algorithms for non-robust problems immediately yield ones for their robust versions.
  • The approximation ratio remains tied to the integrality gap of the original LP.
  • This holds for any discrete problem whose non-robust version admits such a verifier.
  • Robust versions of problems like knapsack or set cover become approximable if their non-robust versions are.

Where Pith is reading between the lines

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

  • The method might inspire similar reductions for other uncertainty models beyond convex objective sets.
  • It could be tested by applying it to specific combinatorial problems and checking if the derived ratio matches known bounds.
  • Connections to parametric optimization or sensitivity analysis in linear programming may exist.

Load-bearing premise

The uncertainty set must be convex to ensure the worst-case objective can be handled through the same LP relaxation verifier.

What would settle it

A counterexample consisting of a specific discrete optimization problem and convex uncertainty set where applying the non-robust integrality gap verifier does not guarantee an approximation for the robust problem.

read the original abstract

We consider robust discrete minimization problems where uncertainty is defined by a convex set in the objective. We show how an integrality gap verifier for the linear programming relaxation of the non-robust version of the problem can be used to derive approximation algorithms for the robust version.

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 considers objective-robust discrete minimization problems in which the uncertainty set on the objective coefficients is convex. It claims that any integrality-gap verifier for the LP relaxation of the corresponding non-robust problem can be used, via a black-box reduction, to obtain approximation algorithms for the robust version.

Significance. If the claimed reductions hold, the result supplies a general, modular technique for lifting existing integrality-gap verifiers to the robust setting without re-deriving problem-specific analyses. This could streamline the construction of approximation algorithms for a range of combinatorial problems under convex objective uncertainty.

minor comments (2)
  1. The abstract states the high-level claim but does not exhibit the reduction or the role of convexity; the manuscript should include a concise statement of the reduction (e.g., as a numbered theorem or algorithm) early in the introduction or in a dedicated section.
  2. Notation for the uncertainty set, the robust objective, and the integrality-gap verifier should be introduced uniformly and used consistently throughout; a table or short glossary of symbols would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our work and for the positive significance assessment. The recommendation of minor revision is noted. However, the report lists no specific major comments, so we have no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims a black-box reduction: an integrality gap verifier for the non-robust LP relaxation yields approximation algorithms for the objective-robust version when uncertainty is convex. The provided abstract and description contain no equations, no fitted parameters renamed as predictions, no self-citations invoked as load-bearing uniqueness theorems, and no ansatzes smuggled via prior work. The reduction is presented as a composition enabled by convexity, with no evidence that any step reduces by construction to its own inputs. This is the normal case of a self-contained algorithmic reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are stated or can be extracted.

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discussion (0)

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Reference graph

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