Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity

Grigoriev, Dima

doi:10.1016/s0304-3975(00)00157-2

Cited by 158 publications

(190 citation statements)

References 16 publications

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“…In [10] was used different notion of expanders but it is easy to see that the result [10,Lemma 8] is also correct in the following form: n/2, d, c)-expander, then the degree of every PS derivation of the system of equalities Tb G is at least 0 n, where Tb G is the binomial representation of the Tseitin formula based on the graph G.…”

Section: Linear Lower Bound On the Boolean Degree Of The Ps Proof Of mentioning

confidence: 99%

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

Kojevnikov

Itsykson

2006

Automata, Languages and Programming

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We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.

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Section: Linear Lower Bound On the Boolean Degree Of The Ps Proof Of mentioning

confidence: 99%

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

Kojevnikov

Itsykson

2006

Automata, Languages and Programming

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“…On the other hand, it was shown by Grigoriev, Hirsch, and Pasechnik [27] that the Tseitin formulas are not hard for a system very related to S(k), for some constant k. Even more, a careful look at their proof shows that any unsolvable system of linear equations mod 2 in which each equation has at most three non-zero coefficients, when appropriately encoded as a set of clauses, has polynomial-size refutations in S(5) by simulating Gaussian elimination. This should be put in contrast with the results of Grigoriev [26] that imply that any SA + k -refutation of the Tseitin formulas requires k = Ω(n), where n is the number of variables.…”

Section: Semi-algebraic Proofs and Linear Equations Modmentioning

confidence: 73%

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Atserias

2013

Theory and Applications of Satisfiability Testing – SAT 2013

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Abstract. In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose proof-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.

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“…Random CSP is extensively studied due to its importance to average-case complexity, probability and statistical physics [MZK + 99], cryptography [ABW10], hardness of approximation [Fei02] and other fields. It is also a prototypical example of a hard CSP [Gri01,Sch08]. Random Label-Cover, i.e., random approximate CSP, is therefore a natural object for study.…”

Section: Label-covermentioning

confidence: 99%

Approximation Algorithms for Label Cover and The Log-Density Threshold

Chlamtáč¹,

Manurangsi²,

Moshkovitz³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L, R, E) and each edge e = (x, y) ∈ E carries a projection π e that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N −c where N = nk, n is the number of vertices, k is the number of labels, and 0 < c < 1 is some constant.Inspired by a framework originally developed for Densest k-Subgraph, we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is N 3−2 √ 2 ≈ N −0.17 . We then design, for any ε > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N 3−2 √ 2+ε . In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. time algorithm whose approximation ratio is roughly N −0.233 . The previous best efficient approximation ratio was N −0.25 . We present some evidence towards an N −c threshold by constructing integrality gaps for N Ω(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N −0.25 , and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N −0.25+ε for any ε > 0.

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Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity

Cited by 158 publications

References 16 publications

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Approximation Algorithms for Label Cover and The Log-Density Threshold

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