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”.