Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems

Abstract

In the Hitting Set problem, we are given a collection F of subsets of a ground set V and an integer p , and asked whether V has a p -element subset that intersects each set in F . We consider two parameterizations of Hitting Set below tight upper bounds, p = m − k and p = n − k . In both cases k is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP ⊆ NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H = ( V , F ) , makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of H = ( V , F ) is the minimum integer d such that for each X ⊂ V the hypergraph with vertex set V ∖ X and edge set containing all edges of F without vertices in X , has a vertex of degree at most d . In Nonblocker ( Directed Nonblocker), we are given an undirected graph (a directed graph) G on n vertices and an integer k , and asked whether G has a set X of n − k vertices such that for each vertex y ∉ X there is an edge (arc) from a vertex in X to y . Nonblocker can be viewed as a special case of Directed Nonblocker (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that Nonblocker has a linear-order kernel. We obtain a linear-order kernel for Directed Nonblocker.