When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.80
URN: urn:nbn:de:0030-drops-73804
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Raghavendra, Prasad ; Weitz, Benjamin

On the Bit Complexity of Sum-of-Squares Proofs

LIPIcs-ICALP-2017-80.pdf (0.5 MB)


It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O'Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs. First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common use-cases of the SoS algorithm, such as Max-CSP, Balanced Separator, Max-Clique, Max-Bisection, and Unit-Vector constraints. On the negative side, O'Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and non-negative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n).

BibTeX - Entry

  author =	{Prasad Raghavendra and Benjamin Weitz},
  title =	{{On the Bit Complexity of Sum-of-Squares Proofs}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{80:1--80:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-73804},
  doi =		{10.4230/LIPIcs.ICALP.2017.80},
  annote =	{Keywords: Sum-of-Squares, Combinatorial Optimization, Proof Complexity}

Keywords: Sum-of-Squares, Combinatorial Optimization, Proof Complexity
Seminar: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 06.07.2017

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