DIMACS TR: 99-62

Dual-Bounded Hypergraphs: Generating Partial and Multiple Transversals

Authors: E. Boros, V. Gurvich, L. Khachiyan and K. Makino


We consider two natural generalizations of the notion of transversal to a finite hypergraph, so called {\em multiple} and {\em partial} transversals. We show that the hypergraphs of all multiple and all partial transversals are dual-bounded in the sense that in both cases, the size of the dual hypergraph is bounded by a polynomial in the cardinality and the length of description of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold Boolean logic, which may be of independent interest. We also show that the problems of generating all multiple and all partial transversals for a given hypergraph are polynomial-time reducible to the generation of all ordinary transversals for another hypergraph, i.e., to the well-known dualization problem for hypergraphs. As a corollary, we obtain incremental quasi-polynomial-time algorithms for both of the above problems, as well as for the generation of all the minimal Boolean solutions for an arbitrary monotone system of linear inequalities.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-62.ps.gz
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