On the hardness of inferring phylogenies from triplet-dissimilarities

This work considers the problem of reconstructing a phylogenetic tree from triplet-dissimilarities, which are dissimilarities defined over taxon-triplets. Triplet-dissimilarities are possibly the simplest generalization of pairwise dissimilarities, and were used for phylogenetic reconstructions in the past few years. We study the hardness of finding a tree best fitting a given triplet-dissimilarity table under the ℓ∞ norm. We show that the corresponding decision problem is NP-hard and that the corresponding optimization problem cannot be approximated in polynomial time within a constant multiplicative factor smaller than 1.4. On the positive side, we present a polynomial time constant-rate approximation algorithm for this problem. We also address the issue of best-fit under maximal distortion, which corresponds to the largest ratio between matching entries in two triplet-dissimilarity tables. We show that it is NP-hard to approximate the corresponding optimization problem within any constant multiplicative factor.