On the Enumeration of Minimal Hitting Sets in Lexicographical Order
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It is a long-standing open problem whether there exists an output-polynomial algorithm enumerating all minimal hitting sets of a hypergraph. A stronger requirement is to ask for an algorithm that outputs them in lexicographical order. We show that there is no incremental-polynomial algorithm for the ordered enumeration, unless P = NP. Notwithstanding, we present a method with delay O(|H|^(k*+2) |V|^2), where k* is the rank of the transversal hypergraph. On classes of hypergraphs for which k* is bounded the delay is polynomial. Additionally, we prove that the extension problem of minimal hitting sets is W[3]-complete when parameterised by the size of the set which is to be extended. For the latter problem, we give an algorithm that is optimal under ETH.
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