The complexity of facets resolved
Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Journal of the ACM (JACM)
Improved approximations of packing and covering problems
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On subclasses of minimal unsatisfiable formulas
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference
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An Application of Matroid Theory to the SAT Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Lean clause-sets: generalizations of minimally unsatisfiable clause-sets
Discrete Applied Mathematics - The renesse issue on satisfiability
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
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Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Incompressibility through Colors and IDs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
On problems without polynomial kernels
Journal of Computer and System Sciences
Solving satisfiability in less than 2n steps
Discrete Applied Mathematics
Theoretical Computer Science
Parameterized Complexity
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We consider a CNF formula F as a multiset of clauses: F={c1,…, cm}. The set of variables of F will be denoted by V(F). Let BF denote the bipartite graph with partite sets V(F) and F and an edge between v∈V(F) and c∈F if v∈c or $\bar{v} \in c$. The matching number ν(F) of F is the size of a maximum matching in BF. In our main result, we prove that the following parameterization of MaxSat is fixed-parameter tractable: Given a formula F, decide whether we can satisfy at least ν(F)+k clauses in F, where k is the parameter. A formula F is called variable-matched if ν(F)=|V(F)|. Let δ(F)=|F|−|V(F)| and δ*(F)= max F′⊆Fδ(F′). Our main result implies fixed-parameter tractability of MaxSat parameterized by δ(F) for variable-matched formulas F; this complements related results of Kullmann (2000) and Szeider (2004) for MaxSat parameterized by δ*(F). To prove our main result, we obtain an O((2e)2kkO(logk) (m+n)O(1))-time algorithm for the following parameterization of the Hitting Set problem: given a collection $\cal C$ of m subsets of a ground set U of n elements, decide whether there is X⊆U such that C∩X≠∅ for each $C\in \cal C$ and |X|≤m−k, where k is the parameter. This improves an algorithm that follows from a kernelization result of Gutin, Jones and Yeo (2011).