We consider the numerical taxonomy problem of fitting a positive distance function D:(S2)→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((logn)(loglogn)) by Ailon and Charikar [2005] who wrote "Determining whether an O(1) approximation can be obtained is a fascinating question".