Parameterized Complexity of MaxSat Above Average
R. Crowston, G. Gutin, M. Jones, V. Raman and S. Saurabh
In: LATIN 2012, Arequipa, Peru(2012).

## Abstract

In {\sc MaxSat}, we are given a CNF formula $F$ with $n$ variables and $m$ clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let $r_1,\ldots, r_m$ be the number of literals in the clauses of $F$. Then ${\rm asat}(F)=\sum_{i=1}^m (1-2^{-r_i})$ is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least ${\rm asat}(F)$ clauses. In the parameterized problem {\sc MaxSat-AA}, we are to decide whether there is a truth assignment satisfying at least ${\rm asat}(F)+k$ clauses, where $k$ is the (nonnegative) parameter. We prove that {\sc MaxSat-AA} is para-NP-complete and thus, {\sc MaxSat-AA} is not fixed-parameter tractable unless P$=$NP. This is in sharp contrast to the similar problem {\sc MaxLin2-AA} which was recently proved to be fixed-parameter tractable by Crowston {\em et al.} (FSTTCS 2011). In fact, we consider a more refined version of {\sc MaxSat-AA}, {\sc Max-$r(n)$-Sat-AA}, where $r_j\le r(n)$ for each $j$. Alon {\em et al.} (SODA 2010) proved that if $r=r(n)$ is a constant, then {\sc Max-$r$-Sat-AA} is fixed-parameter tractable. We prove that {\sc Max-$r(n)$-Sat-AA} is para-NP-complete for $r(n)=\lceil \log n\rceil.$ We also prove that assuming the exponential time hypothesis, {\sc Max-$r(n)$-Sat-AA} is not in XP already for any $r(n)\ge \log \log n +\phi(n)$, where $\phi(n)$ is any unbounded strictly increasing function. This lower bound on $r(n)$ cannot be decreased much further as we prove that {\sc Max-$r(n)$-Sat-AA} is (i) in XP for any $r(n)\le \log \log n - \log \log \log n$ and (ii) fixed-parameter tractable for any $r(n)\le \log \log n - \log \log \log n - \phi(n)$, where $\phi(n)$ is any unbounded strictly increasing function. The proof uses some results on {\sc MaxLin2-AA}.