Abstract
We consider the problem of finding all minimal transversals of a hypergraph \({\mathcal H}\subseteq 2^V\), given by an explicit list of its hyperedges. We give a new decomposition technique for solving the problem with the following advantages: (i) Global parallelism: for certain classes of hypergraphs, e.g. hypergraphs of bounded edge size, and any given integer k, the algorithm outputs k minimal transversals of \({\mathcal H}\) in time bounded by \({\rm polylog}(|V|,|{\mathcal H}|,k)\) assuming \({\rm poly}(|V|,|{\mathcal H}|,k)\) number of processors. Except for the case of graphs, none of the previously known algorithms for solving the same problem exhibit this feature. (ii) Using this technique, we also obtain new results on the complexity of generating minimal transversals for new classes of hypergraphs, namely hypergraphs of bounded dual-conformality, and hypergraphs in which every edge intersects every minimal transversal in a bounded number of vertices.
This research was supported in part by the National Science Foundation (NSF), grant IIS-0118635. The third author is also grateful for the partial support by DIMACS, the NSF’s Center for Discrete Mathematics and Theoretical Computer Science.
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Khachiyan, L., Boros, E., Elbassioni, K., Gurvich, V. (2005). A New Algorithm for the Hypergraph Transversal Problem. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_78
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DOI: https://doi.org/10.1007/11533719_78
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