The Union of Minimal Hitting Sets: Parameterized Combinatorial Bounds and Counting
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.
Keywordsalgorithms parameterization combinatorial inference counting hypergraph transversals
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- 1.Bafna, V., Reinert, K.: Mass spectrometry and computational proteomics. In: Encyclopedia of Genetics, Genomics, Proteomics and Bioinformatics, Wiley, Chichester (2005)Google Scholar
- 6.Chlebik, M., Chlebikova, J.: Crown reductions for the minimum weighted vertex cover problem. ECCC Report 101, to appear in Discrete Appl. Math. (2004)Google Scholar
- 7.Cicalese, F.: Center for Biotechnology, Univ. Bielefeld (personal communication)Google Scholar
- 8.Damaschke, P.: Parameterized enumeration, transversals, and imperfect phylogeny reconstruction, Theoretical Computer Science 351. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 1–12. Springer, Heidelberg (2004)Google Scholar
- 10.Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-hitting set. ECCC Report 073 (2004)Google Scholar