Tight rank lower bounds for the Sherali–Adams proof system

Dantchev, Stefan; Martin, Barnaby; Rhodes, Mark

doi:10.1016/j.tcs.2009.01.002

Cited by 20 publications

(29 citation statements)

References 14 publications

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“…For example the Nullstellensatz maybe viewed as a static version of the Polynomial Calculus and Sherali-Adams as a static version of the Lovász-Schrijver proof system (cf. [GHP02], [Ste09]). …”

Section: This Class Of Predicates Was Defined By Austrin Andmentioning

confidence: 99%

LS+ Lower Bounds from Pairwise Independence

Tulsiani

Worah

2013

2013 IEEE Conference on Computational Complexity

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We consider the complexity of LS+ refutations of unsatisfiable instances of Constraint Satisfaction Problems (k-CSPs) when the underlying predicate supports a pairwise independent distribution on its satisfying assignments. This is the most general condition on the predicates under which the corresponding MAX k-CSP problem is known to be approximation resistant.We show that for random instances of such k-CSPs on n variables, even after Ω(n) rounds of the LS + hierarchy, the integrality gap remains equal to the approximation ratio achieved by a random assignment. In particular, this also shows that LS + refutations for such instances require rank Ω(n). We also show the stronger result that refutations for such instances in the static LS+ proof system requires size exp(Ω(n)).

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Section: This Class Of Predicates Was Defined By Austrin Andmentioning

confidence: 99%

LS+ Lower Bounds from Pairwise Independence

Tulsiani

Worah

2013

2013 IEEE Conference on Computational Complexity

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“…It can be noted that the definition (3) is quite wasteful in that it forces us to represent the proof in a very inefficient way. Other papers in the semialgebraic proof complexity literature, such as [33,37,28], instead define size in terms of the polynomials in isolation, more along the lines of…”

Section: Nullstellensatzmentioning

confidence: 99%

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

et al. 2021

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We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

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“…Let us next switch focus to upper bounds and show that SAR can simulate resolution refutations efficiently in term of size and rank. We remark that a similar simulation is given [DMR09], but since that paper uses a slightly different definition of Sherali-Adams we give a full description of the simulation here for completeness. We start by introducing notation for two polynomial forms which we will use to represent clauses.…”

Section: Size and Rank Upper Bounds For Sherali-adams Refutationsmentioning

confidence: 99%

“…Dantchev et al [DMR09] proved a rank lower bound on SAR refutations of PHP k k−1 . Let us show how this result can be extended to a pigeon-rank lower bound for EPHP k k−1 .…”

Section: Lower Bound On Rank For Sherali-adams Resolutionmentioning

confidence: 99%

Narrow Proofs May Be Maximally Long

Atserias

Lauria

Nordström

2016

ACM Trans. Comput. Logic

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We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n O(w) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w. IntroductionProof complexity studies how hard it is to prove that propositional logic formulas are tautologies. While the original motivation for this line of research, as discussed in [CR79], was to prove superpolynomial lower bounds on proof size for increasingly stronger proof systems as a way towards establishing NP = co-NP (and hence P = NP), it is probably fair to say that most current research in proof complexity is driven by other concerns.One such concern is the connection to SAT solving. By a standard transformation any propositional logic formula can be converted to another formula in conjunctive normal form (CNF) that has the same size up to constant factors and is unsatisfiable if and only if the original formula is a tautology. Any algorithm for solving SAT defines a proof system in the sense that the execution trace of the algorithm constitutes a polynomial-time verifiable witness of unsatisfiability. 1 In fact, most modern-day SAT solvers can be seen to search for proofs in systems at fairly low levels in the proof complexity hierarchy, and upper and lower bounds for these proof systems hence give information about the potential and limitations of the corresponding SAT solvers. In this work, we focus on such proof systems. BackgroundThe dominant strategy in applied SAT solving today is so-called conflict-driven clause learning (CDCL) [BS97, MS99, MMZ + 01], which is ultimately based on the resolution proof system [Bla37]. The most studied complexity measure for resolution is size (also referred to as * This is the full-length version of the paper [ALN14], which appeared in Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC '14).1 Such a witness is often referred to as a refutation rather than a proof , and these two terms are sometimes used interchangeably.

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Tight rank lower bounds for the Sherali–Adams proof system

Cited by 20 publications

References 14 publications

LS+ Lower Bounds from Pairwise Independence

LS+ Lower Bounds from Pairwise Independence

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Narrow Proofs May Be Maximally Long

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