A New Bound for $3$-Satisfiable MaxSat and its Algorithmic Application
G. Gutin, M. Jones and A. Yeo
In: FCT 2011, Oslo, Norway(2011).

## Abstract

Let $F$ be a CNF formula with $n$ variables and $m$ clauses. $F$ is \emph{$t$-satisfiable} if for any $t$ clauses in $F$, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least $\frac{2}{3}$ of its clauses can be satisfied by a truth assignment. Yannakakis's proof utilizes the fact that $\frac{2}{3}m$ is a lower bound on the expected number of clauses satisfied by a random truth assignment over a certain distribution. A CNF formula $F$ is called \emph{expanding} if for every subset $X$ of the variables of $F$, the number of clauses containing variables of $X$ is not smaller than $|X|.$ In this paper we strengthen the $\frac{2}{3}m$ bound for expanding 3-satisfiable CNF formulas by showing that for every such formula $F$ at least $\frac{2}{3}m + \rho n$ clauses of $F$ can be satisfied by a truth assignment, where $\rho(>0.0019)$ is a constant. Our proof uses a probabilistic method with a sophisticated distribution for truth values. We use the bound $\frac{2}{3}m + \rho n$ and results on matching autarkies to obtain a new lower bound on the maximum number of clauses that can be satisfied by a truth assignment in any 3-satisfiable CNF formula. We use our results above to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In {\sc $3$-S-MaxSat-AE}, we are given a $3$-satisfiable CNF formula $F$ with $m$ clauses and asked to determine whether there is an assignment which satisfies at least $\frac{2}{3}m + k$ clauses, where $k$ is the parameter. Note that Mahajan and Raman (1999) asked whether {\sc $2$-S-MaxSat-AE}, the corresponding problem for $2$-satisfiable formulas, is fixed-parameter tractable. Crowston and the authors of this paper proved in \cite{CroGutJonYeo} that {\sc $2$-S-MaxSat-AE} is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. {\sc $2$-S-MaxSat-AE} appears to be easier than {\sc $3$-S-MaxSat-AE} and, unlike this paper, \cite{CroGutJonYeo} uses only deterministic combinatorial arguments.