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

Authors:
Peter Damaschke;Leonid Molokov
Affiliations:
Department of Computer Science and Engineering, Chalmers University, 41296 Göteborg, Sweden;Department of Computer Science and Engineering, Chalmers University, 41296 Göteborg, Sweden
Venue:
Journal of Discrete Algorithms
Year:
2009

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Abstract

A k-hitting set in a hypergraph is a set of at most k vertices that intersects all hyperedges. We study the union of all inclusion-minimal k-hitting sets in hypergraphs of rank r (where the rank is the maximum size of hyperedges). We show that this union is relevant for certain combinatorial inference problems and give worst-case bounds on its size, depending on r and k. For r=2 our result is tight, and for each r=3 we have an asymptotically optimal bound and make progress regarding the constant factor. The exact worst-case size for r=3 remains an open problem. We also propose an algorithm for counting all k-hitting sets in hypergraphs of rank r. Its asymptotic runtime matches the best one known for the much more special problem of finding one k-hitting set. The results are used for efficient counting of k-hitting sets that contain any particular vertex.