The union of minimal hitting sets: Parameterized combinatorial bounds and counting

Paper in proceeding, 2007

We study how many vertices in a rank-r hypergraph can belong to the union of all inclusion-minimal hitting sets of at most k vertices. This union is interesting in certain combinatorial inference problems with hitting sets as hypotheses, as it provides a problem kernel for likelihood computations (which are essentially counting problems) and contains the most likely elements of hypotheses. We give worst-case bounds on the size of the union, depending on parameters r,k and the size of a minimum hitting set. Our result for r=2 is tight. The exact worst-case size for any r>2 remains widely open. By several hypergraph decompositions we achieve nontrivial bounds with potential for further improvements.