The maximum agreement of two nested phylogenetic networks

Abstract

Given a set N of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork contained in every Ni ∈ N with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In this paper, we prove that the general case of MASN is NP-hard already for two phylogenetic networks, but that the problem can be solved efficiently if the two given phylogenetic networks exhibit a nested structure. We first show that the total number of nodes |V(N)| in any nested phylogenetic network N with n leaves and nesting depth d is O(n(d + 1)). We then describe an algorithm for testing if a given phylogenetic network is nested, and if so, determining its nesting depth in O(|V(N)| · (d + 1)) time. Next, we present a polynomial-time algorithm for MASN for two nested phylogenetic networks N 1, N2. Its running time is O(|V(N1)| · |V(N2)| · (d1 + 1) · (d2 + 1)), where d1 and d2 denote the nesting depths of N1 and N2, respectively. In contrast, the previously fastest algorithm for this problem runs in O(|V(N1)| · |V(N2)| 4 f) time, where f ≥ max{d1, d2}. © Springer-Verlag 2004.