Computing phylogenetic roots with bounded degrees and errors is NP-complete

  • Authors:

  • Tatsuie Tsukiji;Zhi-Zhong Chen

  • Affiliations:

  • Department of Information Science, Tokyo Denki University, Hatoyama, Saitama, Japan;Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Saitama, Japan

  • Venue:
  • Theoretical Computer Science - Computing and combinatorics
  • Year:
  • 2006

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Abstract

In this paper we study the computational complexity of the following optimization problem: given a graph G = (V, E), we wish to find a tree T such that (1) the degree of each internal node of T is at least 3 and at most Δ, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E and {{u, v} : u, v are leaves of T and dT (u, v) ≤ k}, is as small as possible, 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.Let sΔ(k) be the size of the largest clique for which an error-free tree T exists. In the course of our proof, we will determine all trees (possibly with degree 2 nodes) that approximate the (sΔ(k) - 1)-clique by errors at most 2.