NP-hard problems in hierarchical-tree clustering
Acta Informatica
SIAM Journal on Discrete Mathematics
European Journal of Combinatorics - Special issue on discrete metric spaces
On the approximability of numerical taxonomy (fitting distances by tree metrics)
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
l∞ -approximation via subdominants
Journal of Mathematical Psychology
Linearly independent split systems
European Journal of Combinatorics
Discrete Applied Mathematics
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In phylogenetic analysis, one searches for phylogenetic trees that reflect observed similarity between a collection of species in question. To this end, one often invokes two simple facts: (i) Any tree is completely determined by the metric it induces on its leaves (which represent the species). (ii) The resulting metrics are characterized by their property of being additive or, in the case of dated rooted trees, ultra-additive. Consequently, searching for additive or ultra-additive metrics A that best approximate the metric D encoding the observed similarities is a standard task in phylogenetic analysis. Remarkably, while there are efficient algorithms for constructing optimal ultra-additive approximations, the problem of finding optimal additive approximations in the l1 or l∞ sense is NP-hard. In the context of the theory of δ-hyperbolic groups, however, good additive approximations A of a metric D were found by Gromov already in 1988 and shown to satisfy the bound ||D - A||∞ ≤ Δ(D)⌈log2(#X - 1)⌉, where Δ(D), the hyperbolicity of D, i.e. the maximum of all expressions of the form D(u, v) + D(x, y) - max(D(u, x) + D(v, y), D(u, y) + D(v, x)) (u, v, x, y ∈ X). Yet, besides some notable exceptions (e.g. Adv. Appl. Math. 27 (2001) 733-767), the potential of Gromov's concept of hyperbolicity is far from being fully explored within the context of phylogenetic analysis. In this paper, we provide the basis for a systematic theory of Δ ultra-additive and Δ additive approximations. In addition, we also explore the average and worst case behavior of Gromov's bound.