One More Occurrence of Variables Makes Satisfiability Jump from Trivial to NP-Complete

Kratochvı́l, Jan; Savický, Petr; Tuza, Źsolt

doi:10.1137/0222015

Cited by 70 publications

(57 citation statements)

References 6 publications

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“…For instance in [9] it is shown that whereas unrestricted k-SAT is NP-complete, for k ≥ 3, it can be solved easily if each clause has size exactly k and no variable occurs in more than f (k) clauses; but it already becomes NP-complete if variables are allowed to occur at most f (k) + 1 times. Here f (k) asymptotically grows as 2 k /(e · k) ; this bound has meanwhile been improved by other authors [10].…”

Section: Introductionmentioning

confidence: 99%

XSAT and NAE-SAT of linear CNF classes

Porschen

Schmidt²,

Speckenmeyer

et al. 2014

Discrete Applied Mathematics

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XSAT and NAE-SAT are important variants of the propositional satisfiability problem (SAT). Both are studied here regarding their computational complexity of linear CNF formulas. We prove that both variants remain NP-complete for (monotone) linear formulas yielding the conclusion that also bicolorability of linear hypergraphs is NP-complete. The reduction used gives rise to the complexity investigation of both variants for several monotone linear subclasses that are parameterized by the size of clauses or by the number of occurrences of variables. In particular cases of these parameter values we are able to verify the NP-completeness of XSAT respectively NAE-SAT; though we cannot provide a complete treatment. Finally we focus on exact linear formulas where clauses intersect pairwise, and for which SAT is known to be polynomial-time solvable [1]. We verify the same assertion for NAE-SAT relying on a result in [2]; whereas we obtain NP-completeness for XSAT of exact linear formulas. The case of uniform clause size k remains open for the latter problem. However, we can provide its polynomial-time behavior for k at most 6.

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Section: Introductionmentioning

confidence: 99%

XSAT and NAE-SAT of linear CNF classes

Porschen

Schmidt²,

Speckenmeyer

et al. 2014

Discrete Applied Mathematics

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“…Tovey [16], however, proved that for 3-CNF formulas with maximum variable degree f (3)+1 = 4 satisfiability is already NP-complete. Kratochvíl, Savický, and Tuza [17] generalised this sudden jump behaviour to general k: For every fixed k ≥ 3, satisfiability of (k, f (k) + 1)-CNF formulas is NP-complete. It may be somewhat intriguing that one can prove such a result, given that we do not even know the values of f (k) for k ≥ 5; but we will see.…”

Section: A Sudden Jump In Complexitymentioning

confidence: 99%

The Lovász Local Lemma and Satisfiability

Gebauer

Moser

Scheder

et al. 2009

Lecture Notes in Computer Science

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Abstract. We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.Several open problems remain, some of which we mention in the concluding section.

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“…Hence (3, s)-SAT is trivial for s ≤ 3, and NP-complete for s ≥ 4. Kratochvíl, Savický and Tuza [11] generalized this result by showing that for every k ≥ 3 there is some integer s = f (k) such that The best known lower bound for f (k), a consequence of Lovász Local Lemma, is due to Kratochvíl, Savický and Tuza [11]. Theorem 1.6.…”

Section: Introductionmentioning

confidence: 99%

Disproof of the neighborhood conjecture with implications to SAT

Gebauer

2012

Combinatorica

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We study a Maker/Breaker game described by Beck. As a result we disprove a conjecture of Beck on positional games, establish a connection between this game and SAT and construct an unsatisfiable k-CNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovász Local Lemma is tight up to a constant factor.The Maker/Breaker game we study is as follows. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F , with Maker going first. Maker's goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2 n−1 then Breaker has a winning strategy. We disprove this conjecture by establishing an n-uniform hypergraph with maximum neighborhood size 3 · 2 n−3 where Maker has a winning strategy. Moreover, we show how to construct an n-uniform hypergraph with maximum degree 2 n−1 n where Maker has a winning strategy. In addition we show that each n-uniform hypergraph with maximum degree at most 2 n−2 en has a proper halving 2-coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture.Finally, we establish a connection between SAT and the Maker/Breaker game we study. We can use this connection to derive new results in SAT. A (k, s

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One More Occurrence of Variables Makes Satisfiability Jump from Trivial to NP-Complete

Cited by 70 publications

References 6 publications

XSAT and NAE-SAT of linear CNF classes

XSAT and NAE-SAT of linear CNF classes

The Lovász Local Lemma and Satisfiability

Disproof of the neighborhood conjecture with implications to SAT

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