Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy

We give the first tight integrality gap for Vertex Cover in the Sherali-Adams SDP system. More precisely, we show that for every > 0, the standard SDP for Vertex Cover that is strengthened with the level-6 Sherali-Adams system has integrality gap 2− . To the best of our knowledge this is the first nontrivial tight integrality gap for the Sherali-Adams SDP hierarchy for a combinatorial problem with hard constraints. For our proof we introduce a new tool to establish Local-Global Discrepancy which uses simple facts from high-dimensional geometry. This allows us to give Sherali-Adams solutions with objective value n(1/2 + o(1)) for graphs with small (2 + o(1)) vector chromatic number. Since such graphs with no linear size independent sets exist, this immediately gives a tight integrality gap for the Sherali-Adams system for superconstant number of tightenings. In order to obtain a Sherali-Adams solution that also satisfies semidefinite conditions, we reduce semidefiniteness to a condition on the Taylor expansion of a reasonably simple function that we are able to establish up to constant-level SDP tightenings. We conjecture that this condition holds even for superconstant levels which would imply that in fact our solution is valid for superconstant level Sherali-Adams SDPs. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems