Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

Fiorini, Samuel; Huynh, Tony; Weltge, Stefan

doi:10.48550/arxiv.1711.01358

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…since w i + w 1 ≥ 1 for all i = 1, so one has w i ≥ 1 − w 1 > 1/2. Thus, we can assume that (7) has the form (9). Note that inequality…”

Section: T |mentioning

confidence: 99%

“…It is an open problem whether an extended relaxation with this properties exists. Recent results showed the existence [1] and gave an explicit construction [7] of a linear relaxation for MinKnap of quasi-polynomial size with integrality gap 2 + ǫ. This is obtained by giving an approximate formulation for Knapsack Cover inequalities (KC) (see [4] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Bounded Pitch Inequalities for the Min-Knapsack Polytope

Faenza

Malinović

Mastrolilli

et al. 2018

Lecture Notes in Computer Science

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In the min-knapsack problem one aims at choosing a set of objects with minimum total cost and total profit above a given threshold. In this paper, we study a class of valid inequalities for min-knapsack known as bounded pitch inequalities, which generalize the well-known unweighted cover inequalities. While separating over pitch-1 inequalities is NPhard, we show that approximate separation over the set of pitch-1 and pitch-2 inequalities can be done in polynomial time. We also investigate integrality gaps of linear relaxations for min-knapsack when these inequalities are added. Among other results, we show that, for any fixed t, the t-th CG closure of the natural linear relaxation has the unbounded integrality gap.

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“…since w i + w 1 ≥ 1 for all i = 1, so one has w i ≥ 1 − w 1 > 1/2. Thus, we can assume that (7) has the form (9). Note that inequality…”

Section: T |mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Bounded Pitch Inequalities for the Min-Knapsack Polytope

Faenza

Malinović

Mastrolilli

et al. 2018

Lecture Notes in Computer Science

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show abstract

“…It is interesting to observe that there is indeed a gap between the existence of certain extended formulations, and the fact that we can construct them efficiently: for instance in [5], it is shown that there is a small extended formulation for the stable set polytope that is O( √ n)-approximated (with n being the number of nodes of the graph), but we do not expect to obtain it efficiently because of known hardness results [18]. In another case, a proof of the existence of a subexponential formulation with integrality gap 2 + ǫ for min-knapsack [4] predated the efficient construction of a formulation with these properties [12].…”

Section: Introductionmentioning

confidence: 99%

Extended Formulations from Communication Protocols in Output-Efficient Time

Aprile

Faenza

2019

Integer Programming and Combinatorial Optimization

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Deterministic protocols are well-known tools to obtain extended formulations, with many applications to polytopes arising in combinatorial optimization. Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the number of rows of the slack matrix (hence, on the exact description in the original space). We give general sufficient conditions under which those tools can be implemented as to be output-efficient, showing applications to e.g. Yannakakis' extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known. For specific classes of polytopes, we give also a direct, efficient construction of extended formulations arising from protocols. Finally, we deal with extended formulations coming from unambiguous non-deterministic protocols.

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“…We now use the term notch to avoid confusion with the definition of pitch due to Bienstock & Zuckerberg [1]. The difference between pitch and notch is discussed in [11].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Polytopes in the 0/1-Cube with Bounded Chvátal-Gomory Rank

et al. 2018

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Let S ⊆ {0, 1} n and R be any polytope contained in [0, 1] n with R ∩ {0, 1} n = S. We prove that R has bounded Chvátal-Gomory rank (CG-rank) provided that S has bounded notch and bounded gap, where the notch is the minimum integer p such that all p-dimensional faces of the 0/1-cube have a nonempty intersection with S, and the gap is a measure of the size of the facet coefficients of conv(S).Let H[S] denote the subgraph of the n-cube induced by the vertices not in S. We prove that if H[S] does not contain a subdivision of a large complete graph, then both the notch and the gap are bounded. By our main result, this implies that the CG-rank of R is bounded as a function of the treewidth of H [S]. We also prove that if S has notch 3, then the CG-rank of R is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee [7], who proved that the CG-rank is bounded by a constant if the treewidth of H[S] is at most 2.

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Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

Cited by 4 publications

References 17 publications

On Bounded Pitch Inequalities for the Min-Knapsack Polytope

On Bounded Pitch Inequalities for the Min-Knapsack Polytope

Extended Formulations from Communication Protocols in Output-Efficient Time

Characterizing Polytopes in the 0/1-Cube with Bounded Chvátal-Gomory Rank

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