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$\mathcal{D}:\binom{S}{2}\rightarrow \mathbb{R}_{> 0}$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$\mathcal{D}$. A nice application is in evolutionary biology where the tree$T$aims to approximate the branching process leading to the observed distances in$\mathcal{D}$[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$O((\log n)(\log\log n)$) by Ailon and Charikar [2005] who wrote “Determining whether an$O(1)$approximation can be obtained is a fascinating question”.