Finding Maximum Common Contractions Between Phylogenetic Networks

TL;DR

This paper explores the comparison of phylogenetic networks using edge contractions and expansions, establishing a distance measure, analyzing computational complexity, and providing efficient algorithms for specific network classes.

Contribution

It introduces a formal framework for network comparison, proves NP-hardness of maximum common contraction, and offers a polynomial-time solution for weakly-galled trees.

Findings

Edge contractions and expansions connect all phylogenetic networks on the same leaves.

Computing maximum common contraction is NP-hard in general.

Polynomial-time algorithm exists for weakly-galled trees.

Abstract

In this paper, we lay the groundwork on the comparison of phylogenetic networks based on edge contractions and expansions as edit operations, as originally proposed by Robinson and Foulds to compare trees. We prove that these operations connect the space of all phylogenetic networks on the same set of leaves, even if we forbid contractions that create cycles. This allows to define an operational distance on this space, as the minimum number of contractions and expansions required to transform one network into another. We highlight the difference between this distance and the computation of the maximum common contraction between two networks. Given its ability to outline a common structure between them, which can provide valuable biological insights, we study the algorithmic aspects of the latter. We first prove that computing a maximum common contraction between two networks is NP-hard, even when the maximum degree, the size of the common contraction, or the number of leaves is bounded. We also provide lower bounds to the problem based on the Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time algorithm for weakly-galled trees, a generalization of galled trees.