How Tight is the Corner Relaxation? Insights Gained from the Stable Set Problem

Cornuéjols, Gérard; Michini, Carla; Nannicini, Giacomo

doi:10.21236/ada586516

Cited by 4 publications

(5 citation statements)

References 14 publications

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“…One inspiration for this is how Gomory's finite cutting-plane algorithm makes progress by incrementally cutting off the optimal face [2,210,1]. This may also help with obtaining stronger cuts; for example, the paper [94] shows how different corner relaxations for the Stable Set problem have very different strengths. Moreover, it is known that the split closure can be obtained from generating split cuts from multiple bases [11] (see [34,100,126] for computational uses of this idea).…”

Section: Cutting-plane Selection: the Need To Ask Different Questionsmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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While many classes of cutting-planes are at the disposal of integer programming solvers, our scientific understanding is far from complete with regards to cutting-plane selection, i.e., the task of selecting a portfolio of cutting-planes to be added to the LP relaxation at a given node of the branch-and-bound tree. In this paper we review the different classes of cutting-planes available, known theoretical results about their relative strength, important issues pertaining to cut selection, and discuss some possible new directions to be pursued in order to accomplish cutting-plane selection in a more principled manner. Finally, we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection. * santanu.dey@isye.gatech.edu † molinaro@inf.puc-rio.brIn particular, general-purpose cutting-planes were seen as mostly of theoretical interest. However, in the mid 90s a breakthrough came with a series of papers by Balas, Ceria, Cornuéjols, and Natraj, where they obtained striking computational results using Gomory Mixed Integer cuts and the then newly discovered Lift-and-Project cuts, within the branch-and-bound framework [29,30]. These papers show the efficacy of general-purpose cutting-planes, when properly selected and employed.Today, even more families of cutting-planes are known. While we do not attempt to survey the vast literature on cutting-planes, we discuss the broad perspectives from which they are derived, with some representative examples.and therefore valid inequalities for P \ (int(Q) × R q ) are valid for P ∩ (Z n × R q ). We call such cuts geometric cuts. A set Q is a maximal lattice-free convex set if it is lattice-free, convex, and no other lattice-free convex set properly contains it. A beautiful structural result is that all maximal lattice-free convex sets are polyhedral [169,38]. Note that when Q is a polyhedral} one can generate a geometric cut by ensuring its validity for all the disjunctions identified with the facets of Q, i.e.,Famous geometric cuts include the Chvátal-Gomory (CG) cuts [131,65,164,120], implied bounds [145], and more generally split cuts [27,85,34]. For all these cuts, the underlying lattice-free set is the so-called split set: {x ∈ R n | π 0 ≤ π ⊤ x ≤ π 0 + 1}, where π 0 ∈ Z and π ∈ Z n . More general geometric cuts include cuts from maximal lattice-free convex sets [12,62] (see the excellent survey [39]) and the closely related intersection cuts developed by Balas in 1970s [25]. Geometric cuts called multi-branch split cuts [165,99,104] are based on non-convex lattice-free sets obtained as the union of split sets.Structured relaxations. Given a mixed-integer set P ∩ (Z n × R q ), the basic idea is to construct a relaxation of P , say R ⊇ P , and analyze the mixed-integer set R ∩ (Z n × R q ). Valid inequalities for the relaxation then serve as cutting-planes for the original set, and the additional structure of R eases the search for facets or strong inequalities. Two excellent review articles discu...

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Section: Cutting-plane Selection: the Need To Ask Different Questionsmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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“…Interestingly, however, there are some N P-hard ILPs whose GR can be solved in polynomial time. Indeed, Cornuéjols et al [8] show that this is true for ILPs of the form max p T x :…”

Section: Group Relaxationmentioning

confidence: 97%

On the complexity of surrogate and group relaxation for integer linear programs

Dokka¹,

Letchford

Mansoor

2021

Operations Research Letters

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“…If c wu ′ = 0 (see Figure (11)), then it follows that y vv ′′′ = −y uu ′ by considering y uu ′ + y u ′ v + y vu + y vv ′′′ + y u ′ w + y uv ′ . Then in this case we have that y uw = −y uu ′ = y vv ′′′ = y vw , and therefore y g = y e , as desired.…”

Section: Traveling Salesman Polytopementioning

confidence: 99%

“…The Fractional Stable Set polytope was widely studied. In particular, it is known that this polytope is half-integral [1], and that the vertices of this polytope have a nice graph interpretation: namely, they can be mapped to subgraphs of G with all connected components being trees and 1-trees 1 [9,11]. This graphical interpretation of vertices was used in [18] to prove that the combinatorial diameter of the Fractional Stable Set polytope is upper bounded by n. In Section 5, we provide a characterization for circuits of this polytope.…”

Section: Introductionmentioning

confidence: 99%

On the Circuit Diameter of Some Combinatorial Polytopes

Kafer¹,

Pashkovich²,

Sanità³

2019

SIAM J. Discrete Math.

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The combinatorial diameter of a polytope P is the maximum value of a shortest path between two vertices of P , where the path uses the edges of P only. In contrast to the combinatorial diameter, the circuit diameter of P is defined as the maximum value of a shortest path between two vertices of P , where the path uses potential edge directions of P i.e., all edge directions that can arise by translating some of the facets of P .In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Salesman polytope and the Fractional Stable Set polytope.

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How Tight is the Corner Relaxation? Insights Gained from the Stable Set Problem

Cited by 4 publications

References 14 publications

Theoretical challenges towards cutting-plane selection

Theoretical challenges towards cutting-plane selection

On the complexity of surrogate and group relaxation for integer linear programs

On the Circuit Diameter of Some Combinatorial Polytopes

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