On bounded pitch inequalities for the min-knapsack polytope

Faenza, Yuri; Malinović, Igor; Mastrolilli, Monaldo; Svensson, Ola

doi:10.48550/arxiv.1801.08850

Cited by 1 publication

(2 citation statements)

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“…Given remark (4) one wonders if a result similar to Theorem 2 exists for the minimum-knapsack problem. This question has been taken up in recent work [5], where the following result is proved:…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 5 [5] Given a minimum-knapsack problem (2), 0 < ǫ < 1 and p = 1 or p = 2, there is an algorithm that, with input x * ∈ [0, 1] n either finds a valid inequality for (2) of pitch = p that is violated by x * or shows that x * satisfies all valid inequalities of pitch p within multiplicative error ǫ. The complexity of the algorithm is polynomial in n and 1/ǫ.…”

Section: Introductionmentioning

confidence: 99%

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Simpler derivation of bounded pitch inequalities for set covering, and minimum knapsack sets

Bienstock,

Zuckerberg

2018

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A valid inequality α T x ≥ α0 for a set covering problem is said to have pitch ≤ π (π a positive integer) if the π smallest positive αj sum to at least α0. This paper presents a new, simple derivation of a relaxation for set covering problems whose solutions satisfy all valid inequalities of pitch ≤ π and is of polynomial size, for each fixed π. We also consider the minimum knapsack problem, and show that for each fixed integer p > 0 and 0 < ǫ < 1 one can separate, within additive tolerance ǫ, from the relaxation defined by the valid inequalities with coefficients in {0, 1, . . . , p} in time polynomial in the number of variables and 1/ǫ.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%