Locating a Tree in a Phylogenetic Network in Quadratic Time

Abstract

International audienceA fundamental problem in the study of phylogenetic networks is to determine whether or not a given phylogenetic network contains a given phylogenetic tree. We develop a quadratic-time algorithm for this problem for binary nearly-stable phylogenetic networks. We also show that the number of reticulations in a reticulation visible or nearly stable phylogenetic network is bounded from above by a function linear in the number of taxa

On Tree-Based Phylogenetic Networks

“…In fact, since a reticulation visible network over X has at most 4(n − 1) reticulation vertexes (Gambette et al, 2015), more than 2 4(n−1) different trees on X cannot all be displayed in a reticulation visible phylogenetic tree simultaneously.…”

“…A network is reticulation visible if every reticulation vertex is visible. Reticulation visible networks are tree-based (Francis and Steel, 2015;Gambette et al, 2015).…”

Phylogenetic Networks that Display a Tree Twice

“…The latter problem is related to calculating the parsimony score of a network [13] which, given the popularity of parsimony tree reconstruction algorithms, is likely to become a standard tool in computing a phylogenetic network directly from sequence data. While deciding if a tree is displayed by a network is polynomial-time solvable for certain special classes of phylogenetic networks (for the work done on the aforementioned problem see [14], [15], [16], [17], [18]), the problem is NP-complete in its general form [19]. Similarly, counting the number of phylogenetic trees that are displayed in an arbitrary phylogenetic network is also known to be a computationally hard problem [20].…”

Solving the Tree Containment Problem for Genetically Stable Networks in Quadratic Time

“…Note that in a nearly tree-child network, there is a tree path from one of the parents of every reticulation to a leaf, so that parent must be stable. On the other hand, a GS network may not be a nearly tree-child network 3 . Therefore, GS networks comprise a proper superclass of the nearly tree-child networks.…”

“…Biologists therefore demand that the network display existing gene trees, and the corresponding algorithmic problem is known as the tree containment problem (or TC problem for short) [5], which is well-known to be NP-complete [7,6]. Great efforts have been devoted to identifying tractable subclasses of networks, such as binary galled trees [7], normal networks, binary tree-child networks, level-k 1 networks [6], or nearly-stable networks [3]. One of the major open questions in this setting is the complexity of the TC problem on the so-called stable networks.…”