On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)

Reviewer: Luigi Gatteschi

This paper is concerned with the so-called numerical taxonomy problem, that is, the problem of fitting an n × n distance matrix d by a tree metric T. Let e be the <__?__Pub Fmt nolinebreak>closest<__?__Pub Fmt /nolinebreak> additive metric under the L ? norm, that is, e = mi n T ? T - D ? ? . The authors present an O n 2 algorithm for finding an additive metric T , such that ? T - D ? ? ? 3 e . Furthermore, they complement the result by not only finding that an L - ? -optimal solution is <__?__Pub Fmt italic>NP<__?__Pub Fmt /italic>-hard, but also showing that it is <__?__Pub Fmt italic>NP<__?__Pub Fmt /italic>-hard to find an additive metric T such that ? T - D ? ? ? 3 8 e . The algorithm presented is achieved by transforming the general tree metric problem to that of ultrametrics, with a loss of a factor of 3 on the approximation ratio. Since the ultrametric problem is optimally solvable, the result follows. A generalization to other norms L k with finite k <__?__Pub Caret>concludes this interesting and well-written paper. It will be fully appreciated by researchers in the area of data structures and their applications.