Some advances on Lovász–Schrijver semidefinite programming relaxations of the fractional stable set polytope

Abstract

We study Lovász and Schrijver’s hierarchy of relaxations based on positive semidefiniteness constraints derived from the fractional stable set polytope. We show that there are graphs $G$ for which a single application of the underlying operator, $N_{+}$, to the fractional stable set polytope gives a nonpolyhedral convex relaxation of the stable set polytope. We also show that none of the current best combinatorial characterizations of these relaxations obtained by a single application of the $N_{+}$ operator is exact.