Parameterized Complexity of Computing Maximum Minimal Blocking and Hitting Sets

Abstract

A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G. Let \(\textsf {mmbs} (G)\) be the size of a maximum (inclusion-wise) minimal blocking set of G. This parameter has recently played an important role in the kernelization of Vertex Cover with structural parameterizations. We provide a panorama of the complexity of computing \(\textsf {mmbs} \) parameterized by the natural parameter and the independence number of the input graph. We also consider the closely related parameter \(\textsf {mmhs} \), which is the size of a maximum minimal hitting set of a hypergraph. Finally, we consider the problem of computing \(\textsf {mmbs} \) parameterized by treewidth, especially relevant in the context of kernelization. Since a blocking set intersects every maximum-sized independent set of a given graph and properties involving counting the sizes of arbitrarily large sets are typically non-expressible in monadic second-order logic, its tractability does not seem to follow from Courcelle’s theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem.