Towards Optimal Integrality Gaps for Hypergraph Vertex Cover in the Lovász-Schrijver Hierarchy

“Lift-and-project” procedures, which tighten linear relaxations over many rounds, yield many of the celebrated approximation algorithms of the past decade or so, even after only a constant number of rounds (e.g., for max-cut, max-3sat and sparsest-cut). Thus proving super-constant round lowerbounds on such procedures may provide evidence about the inapproximability of a problem. We prove an integrality gap of k–e for linear relaxations obtained from the trivial linear relaxation for k-uniform hypergraph vertex cover by applying even Ω(loglog n) rounds of Lovasz and Schrijver's LS lift-and-project procedure. In contrast, known PCP-based results only rule out k–1–e approximations. Our gaps are tight since the trivial linear relaxation gives a k-approximation.