Approximating the Best-Fit Tree Under Lp Norms

We consider the problem of fitting an n × n distance matrix M by a tree metric T . We give a factor O(min{n, (k log n)}) approximation algorithm for finding the closest ultrametric T under the Lp norm, i.e. T minimizes ‖T, M‖p. Here, k is the number of distinct distances in M . Combined with the results of [1], our algorithms imply the same factor approximation for finding the closest tree metric under the same norm. In [1], Agarwala et al. present the first approximation algorithm for this problem under L∞. Ma et al. [2] present approximation algorithms under the Lp norm when the original distances are not allowed to contract and the output is an ultrametric. This paper presents the first algorithms with performance guarantees under Lp (p < ∞) in the general setting. We also consider the problem of finding an ultrametric T that minimizes Lrelative: the sum of the factors by which each input distance is stretched. For the latter problem, we give a factor O(log n) approximation.