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.
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Damaschke, P. (2007). The Union of Minimal Hitting Sets: Parameterized Combinatorial Bounds and Counting. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_29
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DOI: https://doi.org/10.1007/978-3-540-70918-3_29
Publisher Name: Springer, Berlin, Heidelberg
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