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

Abstract

Abstract.A phylogenetic network is a rooted acyclic digraph whose leaves are labeled with a set of taxa. The tree containment problem is a fundamental problem arising from model validation in the study of phylogenetic networks. It asks to determine whether or not a given network displays a given phylogenetic tree over the same leaf set. It is known to be NP-complete in general. Whether or not it remains NP-complete for stable networks is an open problem. We make progress towards answering that question by presenting a quadratic time algorithm to solve the tree containment problem for a new class of networks that we call genetically stable networks, which include tree-child networks and comprise a subclass of stable networks.

Solving the tree containment problem in linear time for nearly stable phylogenetic networks

“…A network is nearly stable if for every vertex, either the vertex or its parents are stable [10]. It is genetically stable if every reticulation vertex is stable and has at least one stable parent [11]. (1) Every genetically stable network is tree-sibling.…”

“…To tackle the TC problem for reticulation-visible networks, we introduced nearly stable networks and genetically stable networks, which both generalize tree-child networks, and gave quadratic time algorithms for solving the TC problem on both classes [10,11]. These results gave an insight on the topological structure of a reticulation-visible network, and eventually led to a solution to the open problem in [12].…”

A decomposition theorem and two algorithms for reticulation-visible networks

“…For each r ∈ IR(C), A r denotes the set of ambiguous leaves defined in Eqn. (12) and lca T C (r) denotes the LCA of the leaves in A r . Proposition 12.…”

“…Because of their importance, a lot of effort has been made to find network classes for which the TCP and CCP are solvable in polynomial time. For example, the TCP has been shown to be solvable in polynomial time for two rather restricted subclasses of reticulationvisible networks [12,26]. Here, the reticulation-visibility property was originally introduced to capture an important feature of galled networks [17].…”

Linear Time Algorithm for Tree-Child Network Containment

“…There are even stronger results for some network classes: deciding whether a tree is contained in a genetically stable network can be done in quadratic time [5], and making this decision for a binary nearly-stable network takes linear time [6]. For the class of tree-child networks, Tree Containment is known to be solvable in linear time [6,7].…”