Multiple Hypernode Hitting Sets and Smallest Two-Cores with Targets

The multiple weighted hitting set problem is to find a subset of nodes in a hypergraph that hits every hyperedge in at least m nodes. We extend the problem to a notion of hypergraphs with so-called hypernodes and show that, for m = 2, it remains fixed-parameter tractable (FPT), parameterized by the number of hyperedges. This is accomplished by a nontrivial extension of the dynamic programming algorithm for hypergraphs. The algorithm might be interesting for certain assignment problems, but here we need it as a tool to solve another problem motivated by network analysis: A d-core of a graph is a subgraph in which every vertex has at least d neighbors. We give an FPT algorithm that computes a smallest 2-core including a given set of target vertices, where the number of targets is the parameter. This FPT result is best possible in the sense that no FPT algorithm for 3-cores can be expected.