Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D P -complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+ k clauses can be recognized in time n O(k) . We imp...

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Bibliographic Details
Published in:Journal of computer and system sciences Vol. 69; no. 4; pp. 656 - 674
Main Author: Szeider, Stefan
Format: Journal Article
Language:English
Published: Elsevier Inc 01.12.2004
Subjects:
Bipartite matching
DP-complete problem
Expansion
Fixed-parameter complexity
Minimal unsatisfiability
SAT problem
Tree-width
Tree-width
Fixed-parameter complexity
SAT problem
Minimal unsatisfiability
Bipartite matching
Expansion
D P -complete problem
ISSN:0022-0000, 1090-2724
Online Access:Get full text
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Summary:Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D P -complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+ k clauses can be recognized in time n O(k) . We improve this result and present an algorithm with time complexity O(2 kn 4) ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time O(2 kn 3) whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails.
ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2004.04.009