Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets

TL;DR

This paper investigates the computational complexity of reconstructing binary level-1 phylogenetic networks from sets of binets and trinets, providing polynomial-time algorithms for certain cases and an exponential algorithm for the general case.

Contribution

It establishes NP-hardness for trinets, polynomial-time solutions for binets and specific trinets, and introduces a general exponential algorithm for reconstructing phylogenetic networks from binets and trinets.

Findings

NP-hardness for trinets reconstruction

Polynomial-time algorithms for binets and certain trinets

An exponential time algorithm for general cases

Abstract

Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level-1 phylogenetic network displaying a given set $\mathcal{T}$ of binary binets or trinets over a set $X$ of taxa, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an $O(3^{|X|} poly(|X|))$ time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted phylogenetic networks.