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Lower Bounds for CSP Refutation by SDP Hierarchies

Authors Ryuhei Mori, David Witmer

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Ryuhei Mori
David Witmer

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Ryuhei Mori and David Witmer. Lower Bounds for CSP Refutation by SDP Hierarchies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 41:1-41:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.41

Abstract

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 >> n^{cmplx(P)/2} can be refuted efficiently using an SDP.

In this work, we give evidence that n^{cmplx(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 Lovasz-Schrijver_+ SDP hierarchies cannot refute a random instance of CSP(P) in polynomial time for any m <= n^{t/2-epsilon}.

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Keywords
  • constraint satisfaction problems
  • LP and SDP relaxations
  • average-case complexity

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