Solving MAX-r-SAT above a tight lower bound

We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2^O(k2) 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 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(k²) 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(k²), 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 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 MAX-r-SAT can be extended to a family of Boolean Constraint Satisfaction Problems.