A semidefinite programming relaxation for the generalized stable set problem

Tetsuya Fujie, Akihisa Tamura

Research output: Contribution to journalArticle

Abstract

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grötschel, Lovász and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a lineal' function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semideflnite programming relaxation of Lovász and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.

Original languageEnglish
Pages (from-to)1122-1128
Number of pages7
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
VolumeE88-A
Issue number5
DOIs
Publication statusPublished - 2005

Fingerprint

Semidefinite Programming Relaxation
Stable Set
Convex Sets
Polynomials
Polynomial time
Perfect Graphs
Polytope
Programming
If and only if
Imply
Generalise
Graph in graph theory

Keywords

  • Bidirecled graphs
  • Integer programming
  • Perfect graphs
  • Semidefinite programming

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Hardware and Architecture
  • Information Systems

Cite this

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AB - In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grötschel, Lovász and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a lineal' function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semideflnite programming relaxation of Lovász and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.

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