Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász–Schrijver Hierarchy

Abstract

Abstract

Vertex Cover Resists SDPs Tightened by Local Hypermetric Inequalities

“…A result related to Theorem 1 was proved by Georgiou et al in [10]. The main result of that paper showed that SDP relaxations obtained by tightening the standard linear programming relaxation for vertex cover using O( log n/ log log n) rounds of the LS + "lift-and-project" method of Lovász and Schrijver [18] have integrality gap 2 − o(1).…”

“…We then find a hypermetric inequality supported on all points of S that is nevertheless not valid for any vertex cover SDPs in the Lovász Schrijver hierarchy. In particular, the integrality gaps proved in [10] did not preclude the possibility that adding such concrete constraints as, say, all hypermetric inequalities on 7-points (e.g., the "heptagonal" inequalities) may result in a non-trivial SDP relaxation.…”

“…The main result of that paper showed that SDP relaxations obtained by tightening the standard linear programming relaxation for vertex cover using O( log n/ log log n) rounds of the LS + "lift-and-project" method of Lovász and Schrijver [18] have integrality gap 2 − o(1). The SDPs considered in [10] seem intimately related to those obtained by adding local ℓ 1 or hypermetric constraints. However, the vertex cover SDP relaxation obtained after k rounds of the LS + method is incomparable to the relaxation obtained by adding all order-k hypermetric inequalities to SDP (1).…”

“…Several recent papers [17,12,4,14,13,10] examine whether semidefinite programming (SDP) relaxations of vertex cover might yield better approximations. Goemans and Williamson [11] introduced semidefinite programming relaxations as an algorithmic technique to obtain a 0.878 approximation for max-cut.…”

“…We use the same graph family as in [17,4,13,10]. The SDP solution can be though of as an ℓ 1 metric to which a small perturbation was applied.…”