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

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.