A Semidefinite Programming Relaxation for the Generalized Stable Set Problem

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lov asz and Schrijver to the generalized stable set problem. We de ne a convex set which serves as a relaxation problem, and show that optimizing a linear 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 de nition of the convex set is based on a semide nite programming relaxation of Lov asz and Schrijver for the maximum weight stable set problem, and the equivalent representation using in nitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof. Tetsuya Fujie Department of Management Science, Kobe University of Commerce, Kobe 651-2197, Japan. Phone : +81-78-794-6161, FAX : +81-78-794-6166, e-mail : [email protected] Akihisa Tamura Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. Phone : +81-75-753-7236, FAX : +81-75-753-7272, e-mail : [email protected] This work is supported by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.