Characterizing path-length matrices of unrooted binary trees

Daniele Catanzaro, Raffaele Pesenti and Roberto Ronco
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Daniele Catanzaro: Université catholique de Louvain, LIDAM/CORE, Belgium

No 2024028, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: We extend some recent results on the necessary and sufficient conditions that a symmetric integer matrix of order n ≥3 must satisfy to encode the Path-Length Matrix (PLM) of a Unrooted Binary Tree (UBT) with n leaves. This problem is at the core of the combinatorics of the Balanced Minimum Evolution Problem, a NP-hard problem much studied in the literature on molecular phylogenetics. We show that, for any natural 3 ≤n ≤11, a reduced set of known conditions, excluding Buneman’ strong four-point conditions, is both necessary and sufficient to characterize PLMs of UBTs. In addition, we present a second and more general characterization based solely on linear conditions derived from the topological properties of UBTs.

Keywords: Combinatorial optimization; tree realization; balanced minimum evolution; unrooted binary trees; path-length matrices; Kraft conditions; Buneman’s four-point conditions (search for similar items in EconPapers)
Pages: 27
Date: 2024-12-02
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