Fixed-parameter tractability of satisfying beyond the number of variables

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 $v \in V(F)$ and $c \in F$ if $v \in c$ 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 {\sc MaxSat} (denoted by $(ν(F)+k)$-\textsc{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 $δ^*(F)=\max_{F'\subseteq F} δ(F').$ Our main result implies fixed-parameter tractability of {\sc MaxSat} parameterized by $δ(F)$ for variable-matched formulas $F$; this complements related results of Kullmann (2000) and Szeider (2004) for {\sc MaxSat} parameterized by $δ^*(F)$. To obtain our main result, we reduce $(ν(F)+k)$-\textsc{SAT} into the following parameterization of the {\sc Hitting Set} problem (denoted by $(m-k)$-{\sc Hitting Set}): given a collection $\cal C$ of $m$ subsets of a ground set $U$ of $n$ elements, decide whether there is $X\subseteq U$ such that $C\cap X\neq \emptyset$ for each $C\in \cal C$ and $|X|\le m-k,$ where $k$ is the parameter. Gutin, Jones and Yeo (2011) proved that $(m-k)$-{\sc Hitting Set} is fixed-parameter tractable by obtaining an exponential kernel for the problem. We obtain two algorithms for $(m-k)$-{\sc Hitting Set}: a deterministic algorithm of runtime $O((2e)^{2k+O(\log^2 k)} (m+n)^{O(1)})$ and a randomized algorithm of expected runtime $O(8^{k+O(\sqrt{k})} (m+n)^{O(1)})$. Our deterministic algorithm improves an algorithm that follows from the kernelization result of Gutin, Jones and Yeo (2011).

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