### 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 |
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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 Jan 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics