A New Facet Generating Procedure for the Stable Set Polytope

Xavier, Álinson S.; Campêlo, Manoel

doi:10.1016/j.endm.2011.05.032

Cited by 8 publications

(14 citation statements)

References 7 publications

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“…Then, The second result establishes sufficient conditions for the generated inequalities to be facet defining. These conditions are slightly weaker than those in [23] due to two differences. First, the subsets W 3 , .…”

Section: On the Strength Of Lemmamentioning

confidence: 77%

“…We shall introduce two procedures. The first one is a simple procedure directly based on Lemma 3, whereas the second one is a strenghtening based on the results in [23]. In both procedures, the generation of a valid inequality consists of the following two steps.…”

Section: Two Procedures For Generating Valid Inequalitiesmentioning

confidence: 99%

“…To show (ii ), we still need the following property of the subgraph of G r induced by V 0 r and a certain subset of vertices. Notation ∼ = denotes the affine isomorphism relation [23].…”

Section: Lemma 11mentioning

confidence: 99%

“…Differently from [23], we determine W r+1 in G r (instead of defining an auxiliary graph). For this purpose, we use the following property of G r due to conditions (I)-(II) and (IV).…”

mentioning

confidence: 99%

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General cut-generating procedures for the stable set polytope

Corrêa

Donne

Koch

et al. 2018

Discrete Applied Mathematics

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We propose general separation procedures for generating cuts for the stable set polytope, inspired by a procedure by Rossi and Smriglio and applying a lifting method by Xavier and Campêlo. In contrast to existing cut-generating procedures, ours generate both rank and non-rank valid inequalities, hence they are of a more general nature than existing methods. This is accomplished by iteratively solving a lifting problem, which consists of a maximum weighted stable set problem on a smaller graph. Computational experience on DIMACS benchmark instances shows that the proposed approach may be a useful tool for generating cuts for the stable set polytope. 1 This work has been partially supported by the Stic/AmSud joint program by CAPES (Brazil), CNRS and MAE (France), CONICYT (Chile) and MINCYT (Argentina) -project 13STIC-05-and the Pronem program by FUNCAP/CNPq (Brazil) -project ParGO.

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Section: On the Strength Of Lemmamentioning

confidence: 77%

Section: Two Procedures For Generating Valid Inequalitiesmentioning

confidence: 99%

“…To show (ii ), we still need the following property of the subgraph of G r induced by V 0 r and a certain subset of vertices. Notation ∼ = denotes the affine isomorphism relation [23].…”

Section: Lemma 11mentioning

confidence: 99%

“…Differently from [23], we determine W r+1 in G r (instead of defining an auxiliary graph). For this purpose, we use the following property of G r due to conditions (I)-(II) and (IV).…”

mentioning

confidence: 99%

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General cut-generating procedures for the stable set polytope

Corrêa

Donne

Koch

et al. 2018

Discrete Applied Mathematics

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“…prevents such a group assignment to be chosen. The integer programming formulation consists in maximizing (1) over all binary points z of type (7) satisfying (4)-(5) and (12) for the extreme rays of the set defined by (10)- (11). A final remark is in order with respect to this integer programming formulation.…”

Section: Integer Programming Formulationmentioning

confidence: 99%

On the combinatorics of the 2-class classification problem

Corrêa

Donne

Marenco

2019

Discrete Optimization

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A set of points X = X B ∪ X R ⊆ R d is linearly separable if the convex hulls of X B and X R are disjoint, hence there exists a hyperplane separating X B from X R . Such a hyperplane provides a method for classifying new points, according to which side of the hyperplane the new points lie. When such a linear separation is not possible, it may still be possible to partition X B and X R into prespecified numbers of groups, in such a way that every group from X B is linearly separable from every group from X R . We may also discard some points as outliers, and seek to minimize the number of outliers necessary to find such a partition. Based on these ideas, Bertsimas and Shioda proposed the classification and regression by integer optimization (CRIO) method in 2007. In this work we explore the integer programming aspects of the classification part of CRIO, in particular theoretical properties of the associated formulation. We are able to find facet-inducing inequalities coming from the stable set polytope, hence showing that this classification problem has exploitable combinatorial properties.

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A new lifting theorem for vertex packing

Marı́n

2018

Optim Lett

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A New Facet Generating Procedure for the Stable Set Polytope

Cited by 8 publications

References 7 publications

General cut-generating procedures for the stable set polytope

General cut-generating procedures for the stable set polytope

On the combinatorics of the 2-class classification problem

A new lifting theorem for vertex packing

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