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
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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 -