arXiv - CS - Computational Complexity Pub Date : 2018-05-03 , DOI: arxiv-1805.01310
Thomas Bläsius; Tobias Friedrich; Julius Lischeid; Kitty Meeks; Martin Schirneck

We devise a method to enumerate the inclusion-wise minimal hitting sets of a hypergraph. The algorithm has delay $O( m^{k^*+1} \, n^2)$ on $n$-vertex, $m$-edge hypergraphs, where $k^*$ is the rank of the transversal hypergraph, i.e., the cardinality of the largest minimal solution. In particular, on classes of hypergraphs for which $k^*$ is bounded, the delay is polynomial. The algorithm uses space linear in the input size only. The enumeration methods solves the extension problem for minimal hitting sets as a subroutine. We show that this problem, parameterised by the cardinality of the set which is to be extended, is one of the first natural W[3]-complete problems. We give an algorithm for the subroutine that is optimal under the assumption that $W[2] \neq \mathrm{FPT}$ or the exponential time hypothesis (ETH), respectively. Despite the hardness of the extension problem, we provide empirical evidence indicating that the enumeration outperforms its theoretical worst-case guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations. Our analysis suggest that these hypergraphs exhibit structure that allows the subroutine to be fast on average.

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