On the complexity of computing MP distance between binary phylogenetic trees

TL;DR

This paper proves that computing the Maximum Parsimony (MP) distance between binary phylogenetic trees is NP-hard, highlighting computational challenges and providing an ILP approach for small instances.

Contribution

It establishes NP-hardness of MP distance computation for binary trees, extending previous results, and offers an ILP method for exact calculations on small trees.

Findings

MP distance computation is NP-hard for binary trees.

Hardness persists with a bounded number of states, even two.

An ILP formulation can compute MP distance for small trees.

Abstract

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two \emph{binary} phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic trees, and it is particularly relevant given the prominence of binary trees in the phylogenetics literature. As a corollary to the main hardness result we show that computation of MP distance is also hard on binary trees if the number of states available is bounded. In fact, via a different reduction we show that it is hard even if only two states are available. Finally, as a first response to this hardness we give a simple Integer Linear Program (ILP) formulation which is capable of computing the MP distance exactly for small trees (and for larger trees when only a small number of character states are available) and which is used to computationally verify several auxiliary results required by the hardness proofs.