Abstract
We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and 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 logic, which may be of independent interest. We also show that the problems of generating all multiple and all partial transversals of an arbitrary hypergraph are polynomial-time reducible to the well-known dualization problem of 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. Thus, it is unlikely that these problems are NP-hard.
The research of the first two authors was supported in part by the Office of Naval Research (Grant N00014-92-J-1375), the National Science Foundation (Grant DMS 98-06389), and DIMACS. The research of the third author was supported in part by the National Science Foundation (Grant CCR-9618796). The authors are also thankful to József Beck for helpful discussions.
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Boros, E., Gurvich, V., Khachiyan, L., Makino, K. (2000). Generating Partial and Multiple Transversals of a Hypergraph. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_50
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DOI: https://doi.org/10.1007/3-540-45022-X_50
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