The complexity of comparing multiply-labelled trees by extending phylogenetic-tree metrics

TL;DR

This paper investigates the computational complexity of extending phylogenetic tree metrics to multilabeled trees (MUL-trees), revealing NP-completeness results and fixed-parameter tractability for specific cases, with implications for phylogenetics and related problems.

Contribution

It establishes the NP-completeness of extending key phylogenetic metrics to MUL-trees and explores fixed-parameter tractability, advancing understanding of computational challenges in phylogenetics.

Findings

Computing metric extension on MUL-trees is NP-complete for path-difference and Robinson Foulds distances.

Extension of Robinson Foulds distance is fixed parameter tractable.

Maximum agreement subtree distance on multiple MUL-trees is NP-complete, but fixed-parameter tractable in certain cases.

Abstract

A multilabeled tree (or MUL-tree) is a rooted tree in which every leaf is labelled by an element from some set, but in which more than one leaf may be labelled by the same element of that set. In phylogenetics, such trees are used in biogeographical studies, to study the evolution of gene families, and also within approaches to construct phylogenetic networks. A multilabelled tree in which no leaf-labels are repeated is called a phylogenetic tree, and one in which every label is the same is also known as a tree-shape. In this paper, we consider the complexity of computing metrics on MUL-trees that are obtained by extending metrics on phylogenetic trees. In particular, by restricting our attention to tree shapes, we show that computing the metric extension on MUL-trees is NP complete for two well-known metrics on phylogenetic trees, namely, the path-difference and Robinson Foulds distances. We also show that the extension of the Robinson Foulds distance is fixed parameter tractable with respect to the distance parameter. The path distance complexity result allows us to also answer an open problem concerning the complexity of solving the quadratic assignment problem for two matrices that are a Robinson similarity and a Robinson dissimilarity, which we show to be NP-complete. We conclude by considering the maximum agreement subtree (MAST) distance on phylogenetic trees to MUL-trees. Although its extension to MUL-trees can be computed in polynomial time, we show that computing its natural generalization to more than two MUL-trees is NP-complete, although fixed-parameter tractable in the maximum degree when the number of given trees is bounded.