Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

Szeider, Stefan

doi:10.1007/3-540-45071-8_55

Cited by 12 publications

(2 citation statements)

References 30 publications

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“…The algorithm of [5] is based on the fact that a CNF formula F is satisfiable if and only if there is a truth assignment t to at most δ * (F ) variables such that the CNF formula F t is matched, where F t results from F by the usual reduction w.r.t. the assignment t, i.e., by removing clauses satisfied by t and literals which are set to FALSE by t. Recently, the quoted result of Fleischner et al could be improved: Szeider [17] presents an algorithm which decides satisfiability of formulas with n variables and maximum deficiency k in time O(2 k · n 3 ); hence SAT for instances of bounded maximum deficiency is "fixed-parameter tractable" in the sense of Downey and Fellows [3].…”

Section: Introductionmentioning

confidence: 99%

Generalizations of Matched CNF Formulas

Szeider

2005

Annals of Mathematics and Artificial Intelligence

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A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is P 2 -complete, answering a question posed by Kleine Büning and Zhao.

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Section: Introductionmentioning

confidence: 99%

Generalizations of Matched CNF Formulas

Szeider

2005

Annals of Mathematics and Artificial Intelligence

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“…Algorithms were provided for MU(1) and MU(2) with quadratic and cubic complexity in [11,8], respectively, and for general MU(k) with complexity n O(k) in [17,13]. Finally, MU(k) was shown to be decidable in O(2 k n 4 ) time, and thus fixed-parameter tractable with respect to k [25].…”

Section: Introductionmentioning

confidence: 99%

Mup

Huang¹

2005

Proceedings of the 2005 Conference on Asia South Pacific Design Automation - ASP-DAC '05

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After establishing the unsatisfiability of a SAT instance encoding a typical design task, there is a practical need to identify its minimal unsatisfiable subsets, which pinpoint the reasons for the infeasibility of the design. Due to the potentially expensive computation, existing tools for the extraction of unsatisfiable subformulas do not guarantee the minimality of the results. This paper describes a practical algorithm that decides the minimal unsatisfiability of any CNF formula through BDD manipulation. This algorithm has a worse-case complexity that is exponential only in the treewidth of the CNF formula. We provide an empirical evaluation of the algorithm, highlighting its efficiency on a set of hard problems as well as its ability to work with existing subformula extraction tools to achieve optimal results.1 The complexity class D P can be defined as all languages that are the intersection of a language in NP and one in coNP [21,20]. D P -complete problems are both NP-hard and coNP-hard.

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Faster Exact Solving of SAT Formulae with a Low Number of Occurrences per Variable

Wahlström

2005

Theory and Applications of Satisfiability Testing

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Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

Cited by 12 publications

References 30 publications

Generalizations of Matched CNF Formulas

Generalizations of Matched CNF Formulas

Mup

Faster Exact Solving of SAT Formulae with a Low Number of Occurrences per Variable

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