Parameterized exact and approximation algorithms for maximum k-set cover and related satisfiability problems

We study the complexity of several parameterizations for max k-set cover. 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, is W[2]-hard. Here, we settle the multiparameterized complexity of max k-set cover under pairs of parameters as (k,∆) and (k, f), where ∆ = maxi{|Si|} and f = maxi |{j|xi ∈ Sj}|. We also study parameterized approximability of the problem with respect to parameters k and p. Then, we investigate some similar parameterizations of a satisfiability problem max sat-k which is close to max k-set cover. Finally, we sketch an enhancement of the classes of the W[·] hierarchy that seems more appropriate for showing completeness of cardinality constrained W[·]-hard problems.