Rooted maximum agreement supertrees

Abstract

Given a set τ rooted, unordered trees, where each T i ∈ τ is distinctly leaf-labeled by a set Λ(T i) and where the sets Λ(T i) may overlap, the maximum agreement supertree problem(MASP) is to construct a distinctly leaf-labeled tree script Q sign with leaf set Λ(script Q sign) ⊆ ∪ Ti ∈ τ Λ(T i) such that |Λ(script Q sign)| is maximized and for each T i ∈ Τ, the topological restriction of T i to Λ(script Q sign) is isomorphic to the topological restriction of script Q sign to Λ(T i). Let n = |∪ Ti ∈ τ Λ(T i)|, k = |Τ|, and D = max Ti ∈ Τ {deg(T i)}. We first show that MASP with k = 2 can be solved in O(√Dn log (2n/D)) time, which is O(n log n) when D = O(1) and O(n 1.5) when D is unrestricted. We then present an algorithm for MASP with D = 2 whose running time is polynomial if k = O(1). On the other hand, we prove that MASP is NP-hard for any fixed k ≥ 3 when D is unrestricted, and also NP-hard for any fixed D ≥ 2 when k is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomial-time (n/log n)-approximation algorithm for MASP. © 2005 Springer Science+Business Media, Inc.