The Complexity of Inferring a Minimally Resolved Phylogenetic Supertree

A recursive algorithm by Aho, Sagiv, Szymanski, and Ullman [1] forms the basis for many modern rooted supertree methods employed in Phylogenetics. However, as observed by Bryant [4], the tree output by the algorithm of Aho et al. is not always minimal; there may exist other trees which contain fewer nodes yet are still consistent with the input. In this paper, we prove strong polynomial-time inapproximability results for the problem of inferring a minimally resolved supertree from a given consistent set of rooted triplets (MINRS). We also present an exponential-time algorithm for solving MINRS exactly which is based on tree separators. It runs in 2O(n log k) time when every node is required to have at most k children which are internal nodes and where n is the cardinality of the leaf label set of the input trees.

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