Fixed-Parameter Tractability of Satisfying beyond the Number of Variables

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 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 B _F . 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′ ⊆ 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 (2000) and Szeider (2004) for MaxSat parameterized by δ ^*(F).

To prove our main result, we obtain an O((2e)^2k k ^O(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).