The complexity of ultrametric partitions on graphs
Information Processing Letters
On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)
SIAM Journal on Computing
A few logs suffice to build (almost) all trees: part II
Theoretical Computer Science
A few logs suffice to build (almost) all trees (l): part I
Random Structures & Algorithms
Low dimensional embeddings of ultrametrics
European Journal of Combinatorics
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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 @?"~ 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.