Lower Bounds for CSP Refutation by SDP Hierarchies

For a k-ary predicate P , a random instance of CSP(P ) with n variables and m constraints is unsatisfiable with high probability when m ≥ O(n). The natural algorithmic task in this regime is refutation: finding a proof that a given random instance is unsatisfiable. Recent work of Allen et al. suggests that the difficulty of refuting CSP(P ) using an SDP is determined by a parameter cmplx(P ), the smallest t for which there does not exist a t-wise uniform distribution over satisfying assignments to P . In particular they show that random instances of CSP(P ) with m ncmplx(P)/2 can be refuted efficiently using an SDP. In this work, we give evidence that ncmplx(P )/2 constraints are also necessary for refutation using SDPs. Specifically, we show that if P supports a (t − 1)-wise uniform distribution over satisfying assignments, then the Sherali-Adams+ and Lovász-Schrijver+ SDP hierarchies cannot refute a random instance of CSP(P ) in polynomial time for any m ≤ nt/2−ε. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems