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Tight Size-Degree Bounds for Sums-of-Squares Proofs

Authors Massimo Lauria, Jakob Nordström

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Massimo Lauria
Jakob Nordström

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Massimo Lauria and Jakob Nordström. Tight Size-Degree Bounds for Sums-of-Squares Proofs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 448-466, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.448

Abstract

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^Omega(d) for values of d = d(n) from constant all the way up to n^delta for some universal constant delta. This shows that the n^O(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Krajicek '04] and [Dantchev and Riis '03], and then applying a restriction argument as in [Atserias, Müller, and Oliva '13] and [Atserias, Lauria, and Nordstrom '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.
Keywords
  • Proof complexity
  • resolution
  • Lasserre
  • Positivstellensatz
  • sums-of-squares
  • SOS
  • semidefinite programming
  • size
  • degree
  • rank
  • clique
  • lower bound

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