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

Authors:
Peter Damaschke
Affiliations:
School of Computer Science and Engineering, Chalmers University, Göteborg, Sweden
Venue:
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Year:
2007

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Abstract

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 k* of a minimum hitting set. (Note that k ≥ k* is allowed.) Our result for r = 2 is tight. The exact worst-case size for any r ≥ 3 remains widely open. By several hypergraph decompositions we achieve nontrivial bounds with potential for further improvements.