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 $${{n^{{\rm O}{(d)}}}}$$ 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 (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.
Highlights
Let f1, . . . , fs ∈ R[x1, . . . , xn] be real, multivariate polynomials
This proof system was introduced by Grigoriev and Vorobjov [15] as an extension of the Nullstellensatz proof system studied by Beame et al [5], and Grigoriev established SOS degree lower bound for unsatisfiable F2-linear equations [13] and for the knapsack problem [12]
We show that using Lasserre semidefinite programming relaxations to find degree-d sums-of-squares proofs is optimal up to constant factors in the exponent of the running time
Summary
Let f1, . . . , fs ∈ R[x1, . . . , xn] be real, multivariate polynomials. the Positivstellensatz proven in [20, 31] says (as a special case) that the the system of equations f1 = 0, . . . , fs = 0. Our focus in this paper is not on algorithmic questions, but more on sums-ofsquares viewed as a proof system ( referred to in the literature as Positivstellensatz or Lasserre) This proof system was introduced by Grigoriev and Vorobjov [15] as an extension of the Nullstellensatz proof system studied by Beame et al [5], and Grigoriev established SOS degree lower bound for unsatisfiable F2-linear equations [13] ( referred to as the 3-XOR problem when each equation involves at most 3 variables) and for the knapsack problem [12]. We refer the reader to, for instance, the introductory section of [26] for more background on sums-of-squares and connections to hardness of approximation, and to the survey [4] for an in-depth discussion of SOS as an approximation algorithm and the intriguing connections to the so-called Unique Games Conjecture [17]
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