Parameterizing above Guaranteed Values: MaxSat and MaxCut

In this paper we investigate the parametrized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellowss7]. Let G be an arbitrary graph having n vertices and m edges, and let f be an arbitrary CNF formula with m clauses on n variables. We improve Cai and Chen's O(2 2k m) time algorithm for determining if at least k clauses of of a c-CNF formula f can be satissedd4]; our algorithm runs in O(jfj+k 2 k) time for arbitrary formulae and in O(m + k k) time for c-CNF formulae. We also give an algorithm for nding a cut of size at least k; our algorithm runs in O(m + n + k4 k) time. Since it is known that G has a cut of size at least d m 2 e and that there exists an assignment to the variables of f that satisses at least d m 2 e clauses of f, we argue that the standard parametrization of these problems is unsuitable. Non-trivial situations arise only for large parameter values, in which range the xed-parameter tractable algorithms are infeasible. A more meaningful question in the parametrized setting is to ask whether d m 2 e + k clauses can be satissed, or d m 2 e + k edges can be placed in a cut. We show that these problems remain xed-parameter tractable even under this parametrization. Furthermore, for upto logarithmic values of the parameter, our algorithms run in polynomial time. We also discuss the complexity of the parametrized versions of these problems where all but k clauses have to satissed or all but k edges have to be in the cut.