## AbstractWe present an exact algorithm that decides, for every fixed $r \geq 2$ in time $O(m) + 2^{O(k^2)}$ whether a given set of $m$ clauses of size $r$ admits a truth assignment that satisfies at least $((2^r-1)m+k)/2^r$ clauses. Thus \textsc{Max-$r$-Sat} is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound~$(1-2^{-r})m$. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with $O(k^2)$ variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size $O(k^2)$, then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of \textsc{Max-$2$-Sat} with $m$ clauses has at least $3k$ variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least $(3m+k)/4$ clauses. We also outline how the fixed-parameter tractability result on \textsc{Max-$r$-Sat} can be extended to a family of Boolean Constraint Satisfaction Problems.
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