Fixed-Parameter Tractability of Satisfying beyond the Number of Variables

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).