On subclasses of minimal unsatisfiable formulas

Büning, Hans Kleine

doi:10.1016/s0166-218x(00)00245-6

Cited by 44 publications

(30 citation statements)

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“…Furthermore, if each literal of a formula F ∈ MU(k), k ≥ 2, is contained in at least 2 clauses, then F is 2-expanding [17,18]. We extend the various quoted results and pinpoint the importance of the notion of q-expansion for satisfiability decision.…”

Section: Maximum Deficiency and Expansionsupporting

confidence: 57%

“…Moreover, deterministic polynomial time algorithms have been developed for the special cases MU(1) and MU (2), based on the very structure of formulas in the respective classes (Davidov, et al [8] and Kleine Büning [18]). Finally it was shown by Kullmann [19] and by Fleischner, et al [12] that for any fixed k, formulas in MU(k) can be recognized in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

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

Szeider

2003

Lecture Notes in Computer Science

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Section: Maximum Deficiency and Expansionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

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

Szeider

2003

Lecture Notes in Computer Science

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“…to deficiency. Formulas in MU(1) can be recognised efficiently [26,27,28] and they have been shown to be universal in the sense that every unsatisfiable CNF formula F has a G ∈ MU(1) with a homomorphism from G to F [29]. We do not define homomorphisms for CNF formulas here.…”

Section: A Special Class Of Unsatisfiable Cnf Formulas -Mu(1)mentioning

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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“…Kleine Büning [10] showed that, if k is a fixed integer, then the recognition problem with deficiency k is in NP, and conjectured that for fixed k, M U (k) can be solved in polynomial time. Finally the question was completely solved by H. Fleischner, O. Kullmann, S.Szeider in [11].…”

Section: Definition 2 (Subclass With Fixed Deficiency)mentioning

confidence: 99%

“…Consider a propositional formula F in conjunctive normal form (CNF), F is minimal unsatisfiable if and only if the formula is unsatisfiable and any proper subformula is satisfiable. There are many existing work on theoretical results [8,9,10,11,12,13,14] and experimental results [15,16] of minimal unsatisfiability. The class of minimal unsatisfiable formulas is denoted as M U and shown to be D P -complete [9].…”

Section: Introductionmentioning

confidence: 99%

Variable Minimal Unsatisfiability

Chen

Ding

2006

Lecture Notes in Computer Science

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Abstract. In this paper, we present variable minimal unsatisfiability (VMU), which is a generalization of minimal unsatisfiability (MU). A characterization of a VMU formula F is that every variable of F is used in every resolution refutation of F . We show that the class of VMU formulas is D P -complete. For fixed deficiency (the difference of the number of clauses and the number of variables), the VMU formulas can be solved in polynomial time. Furthermore, we investigate more subclasses of VMU formulas. Although the theoretic results on VMU and MU are similar, some observations are shown that the extraction of VMU may be more practical than MU in some cases.

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On subclasses of minimal unsatisfiable formulas

Cited by 44 publications

References 10 publications

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

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

The Lovász Local Lemma and Satisfiability

Variable Minimal Unsatisfiability

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