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Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable. (English) Zbl 1350.92036
Summary: Here we show that, given a set of clusters \(\mathcal C\) on a set of taxa \(\mathcal X\), where \(|\mathcal X|=n\), it is possible to determine in time \(f(k)\cdot\mathrm{poly}(n)\) whether there exists a level-\(\leq k\) network (i.e. a network where each biconnected component has reticulation number at most \(k\)) that represents all the clusters in \(\mathcal C\) in the softwired sense, and if so to construct such a network. This extends a result from our paper with L. van Iersel [“On the elusiveness of clusters”, IEEE/ACM Trans. Comput. Biol. Bioinform. 9, No. 2, 517–534 (2012; doi:10.1109/TCBB.2011.128)] which showed that the problem is polynomial-time solvable for fixed \(k\). By defining “\(k\)-reticulation generators” analogous to “level-\(k\) generators”, we then extend this fixed parameter tractability result to the problem where \(k\) refers not to the level but to the reticulation number of the whole network.

92D15 Problems related to evolution
05C05 Trees
68W05 Nonnumerical algorithms
68R10 Graph theory (including graph drawing) in computer science
Full Text: DOI
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