License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.41
URN: urn:nbn:de:0030-drops-66645
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6664/
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Mori, Ryuhei ; Witmer, David

Lower Bounds for CSP Refutation by SDP Hierarchies

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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}.

BibTeX - Entry

@InProceedings{mori_et_al:LIPIcs:2016:6664,
  author =	{Ryuhei Mori and David Witmer},
  title =	{{Lower Bounds for CSP Refutation by SDP Hierarchies}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{41:1--41:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6664},
  URN =		{urn:nbn:de:0030-drops-66645},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.41},
  annote =	{Keywords: constraint satisfaction problems, LP and SDP relaxations, average-case complexity}
}

Keywords: constraint satisfaction problems, LP and SDP relaxations, average-case complexity
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
Issue Date: 2016
Date of publication: 06.09.2016

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