Solving MAX-r-SAT Above a Tight Lower Bound

Abstract

We present an exact algorithm that decides, for every fixed r≥2 in time $O(m)+2^{O(k^{2})}$ whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2r −1)m+k)/2r 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 et al. (J. Comput. Syst. Sci. 75(2):137–153, 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(9r 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(9r k 2), then there is a truth assignment satisfying the required number of clauses. We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the above-mentioned parameterized Max-r-Sat admits a polynomial-size kernel. 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 a certain polynomial time reduction rule to it, then there is a truth assignment that satisfies at least (3m+k)/4 clauses. We also outline how the fixed-parameter tractability and polynomial-size kernel results on Max-r-Sat can be extended to more general families of Boolean Constraint Satisfaction Problems.