Parameterized Algorithms for the Max K-set Cover and Related Satisfiability Problems Parameterized Algorithms for the Max K-set Cover and Related Satisfiability Problems

Given a family of subsets S = {S1, . . . , Sm} over a set of elements X = {x1, . . . , xn} and an integer p, max k-set cover consists of finding a set T of at most k subsets covering at least p elements. This problem, when parameterized by k, can be easily shown to be W[2]-hard. Here, we settle the parameterized complexity of max k-set cover under several parameters as max{k,∆}, where ∆ = maxi{|Si|}, p and max{k, f}, where f = maxi |{j|xi ∈ Sj}|. We also study parameterized approximability of the problem with respect to parameters k and p. We also study parameterization of a satisfiability problem that is linked to max k-set cover in a sense explained in the paper. Finally, we sketch an enhancement of the classes of the W[·] that seems more appropriate for showing completeness of hard W[·]-problems.