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

Szeider, Stefan

Abstract

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; 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 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.

Citation

Szeider, S. (2004). Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. Journal of Computer and System Sciences, 69(4), 656-674. https://doi.org/10.1016/j.jcss.2004.04.009

• Journal Article Type: Article
• Publication Date: 2004-09
• Deposit Date: Oct 14, 2008
• Publicly Available Date: Oct 14, 2008
• Journal: Journal of Computer and System Sciences
• Print ISSN: 0022-0000
• Electronic ISSN: 1090-2724
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 69
• Issue: 4
• Pages: 656-674
• DOI: https://doi.org/10.1016/j.jcss.2004.04.009
• Keywords: SAT problem, Minimal unsatisfiability, Fixed-parameter complexity, DP-complete problem, Tree-width, Bipartite matching, Expansion.
• Public URL: https://durham-repository.worktribe.com/output/1589753