## Abstract

We consider a CNF formula F as a multiset of clauses: F={c _{1},.,c _{m} }. The set of variables of F will be denoted by V(F). Let B _{F} denote the bipartite graph with partite sets V(F) and F and with an edge between vâ̂̂V(F) and câ̂̂F if vâ̂̂c or \bar{v} \in c. The matching number ν(F) of F is the size of a maximum matching in B _{F} . In our main result, we prove that the following parameterization of MaxSat (denoted by (ν(F)+k)-SAT) 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 ^{â̂ -} (F)=max _{F′âŠ†F} δ(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 (IEEE Conference on Computational Complexity, pp. 116-124, 2000) and Szeider (J. Comput. Syst. Sci. 69(4):656-674, 2004) for MaxSat parameterized by δ ^{â̂ -} (F). To obtain our main result, we reduce (ν(F)+k)-SAT into the following parameterization of the Hitting Set problem (denoted by (m-k)-Hitting Set): given a collection \mathcal{C} of m subsets of a ground set U of n elements, decide whether there is XâŠ†U such that Câ̂

Original language | English |
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Journal | Algorithmica |

Volume | 68 |

Issue number | 3 |

Pages (from-to) | 739-757 |

ISSN | 0178-4617 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |