### 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 language | English |
---|---|

Pages (from-to) | 1122-1128 |

Number of pages | 7 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E88-A |

Issue number | 5 |

DOIs | |

Publication status | Published - 2005 |

### Fingerprint

### Keywords

- Bidirecled graphs
- Integer programming
- Perfect graphs
- Semidefinite programming

### ASJC Scopus subject areas

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

### Cite this

**A semidefinite programming relaxation for the generalized stable set problem.** / Fujie, Tetsuya; Tamura, Akihisa.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E88-A, no. 5, pp. 1122-1128. https://doi.org/10.1093/ietfec/e88-a.5.1122

}

TY - JOUR

T1 - A semidefinite programming relaxation for the generalized stable set problem

AU - Fujie, Tetsuya

AU - Tamura, Akihisa

PY - 2005

Y1 - 2005

N2 - 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.

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.

KW - Bidirecled graphs

KW - Integer programming

KW - Perfect graphs

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=24144454205&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24144454205&partnerID=8YFLogxK

U2 - 10.1093/ietfec/e88-a.5.1122

DO - 10.1093/ietfec/e88-a.5.1122

M3 - Article

AN - SCOPUS:24144454205

VL - E88-A

SP - 1122

EP - 1128

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 5

ER -