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Exploring the relationship between max-cut and stable set relaxations

Giandomenico, M and Letchford, A N (2006) Exploring the relationship between max-cut and stable set relaxations. Mathematical Programming, 106 (1). pp. 159-175. ISSN 0025-5610

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Abstract

The max-cut and stable set problems are two fundamental NP-hard problems in combinatorial optimization. It has been known for a long time that any instance of the stable set problem can be easily transformed into a max-cut instance. Moreover, Laurent, Poljak, Rendl and others have shown that any convex set containing the cut polytope yields, via a suitable projection, a convex set containing the stable set polytope. We review this work, and then extend it in the following ways. We show that the rounded version of certain `positive semidefinite' inequalities for the cut polytope imply, via the same projection, a surprisingly large variety of strong valid inequalities for the stable set polytope. These include the clique, odd hole, odd antihole, web and antiweb inequalities, and various inequalities obtained from these via sequential lifting. We also examine a less general class of inequalities for the cut polytope, which we call odd clique inequalities, and show that they are, in general, much less useful for generating valid inequalities for the stable set polytope. As well as being of theoretical interest, these results have algorithmic implications. In particular, we obtain as a by-product a polynomial-time separation algorithm for a class of inequalities which includes all web inequalities.

Item Type: Article
Journal or Publication Title: Mathematical Programming
Uncontrolled Keywords: max-cut problem ; stable set problem ; polyhedral combinatorics
Subjects: H Social Sciences > HB Economic Theory
Departments: Lancaster University Management School > Management Science
ID Code: 44078
Deposited By: ep_importer_pure
Deposited On: 11 Jul 2011 19:09
Refereed?: Yes
Published?: Published
Last Modified: 09 Apr 2014 22:23
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/44078

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