Level-k Phylogenetic Networks Are Constructable from a Dense Triplet Set in Polynomial Time

Abstract

For a given dense triplet set $\mathcal{T}$, there exist two natural questions [7]: Does there exist any phylogenetic network consistent with $\mathcal{T}$? In case such networks exist, can we find an effective algorithm to construct one? For cases of networks of levels k = 0, 1 or 2, these questions were answered in [1,6,7,8,10] with effective polynomial algorithms. For higher levels k, partial answers were recently obtained in [11] with an $O(|\mathcal{T}|^{k+1})$ time algorithm for simple networks. In this paper, we give a complete answer to the general case, solving a problem proposed in [7]. The main idea of our proof is to use a special property of SN-sets in a level-k network. As a consequence, for any fixed k, we can also find a level-k network with the minimum number of reticulations, if one exists, in polynomial time.