    Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor

We consider the numerical taxonomy problem of fitting a positive distance function D:S 2→ℝ_>0 by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in D. A nice application is in evolutionary biology where the tree T aims to approximate the branching process leading to the observed distances in D [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees, and for the special case of ultrametrics with a root having the same distance to all vertices in S. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O((log n)(loglog n)) by Ailon and Charikar  who wrote "Determining whether an O(1) approximation can be obtained is a fascinating question".

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