Computing phylogenetic roots with bounded degrees and errors is hard

Abstract

The DEGREE-Δ CLOSEST PHYLOGENETIC kTH ROOT PROBLEM (ΔCPRk) is the problem of finding a (phylogenetic) tree T from a given graph G = (V, E) such that (1) the degree of each internal node of T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, |E ⊕ {{u, v} : u, v are leaves of T and dT(u, v) ≤ k}| does not exceed a given number, where dT(u, v) denotes the distance between u and v in tree T. We show that this problem is NP-hard for all fixed constants Δ, k ≥ 3. Our major technical contribution is the determination of all pylogenetic roots that approximate the almost largest cliques. Specifically, let sΔ(k) be the size of a largest clique having a kth phylogenetic root with maximum degree Δ. We determine all the phylogenic kth roots with maximum degree Δ that approximate the (sΔ(k) - 1)-clique within error 2, where we allow the internal nodes of phylogeny to have degree 2. © Springer-Verlag Berlin Heidelberg 2004.