Vertex cover resists SDPs tightened by local hypermetric inequalities

Abstract

We consider the standard semidefinite programming (SDP) relaxation for vertex cover to which all hypermetric inequalities supported on at most k vertices have been added. We show that the integrality gap for such SDPs remains 2∈-∈o(1) as long as . This extends successive results by Kleinberg-Goemans, Charikar and Hatami et al. which analyzed integrality gaps of the standard vertex cover SDP relaxation as well as for SDPs tightened using triangle and pentagonal inequalities. Our result is complementary but incomparable to a recent result by Georgiou et al. proving integrality gaps for vertex cover SDPs in the Lovász-Schrijver hierarchy. One of our contributions is making explicit the difference between the SDPs considered by Georgiou et al. and those analyzed in the current paper. We do this by showing that vertex cover SDPs in the Lovász-Schrijver hierarchy fail to satisfy any hypermetric constraints supported on independent sets of the input graph. © 2008 Springer-Verlag Berlin Heidelberg.