A linear bound on the number of states in optimal convex characters for maximum parsimony distance

TL;DR

This paper establishes a linear upper bound on the number of states needed in optimal convex characters for maximum parsimony distance between binary phylogenetic trees, which could improve computational methods in phylogenetics.

Contribution

It proves the first non-trivial linear bound on the number of states in optimal convex characters for maximum parsimony distance in binary trees.

Findings

Bound of at most 7d_MP - 5 states for optimal convex characters

Convex characters with bounded states can be enumerated efficiently

First non-trivial bound on states in optimal characters

Abstract

Given two phylogenetic trees on the same set of taxa X, the maximum parsimony distance d_MP is defined as the maximum, ranging over all characters c on X, of the absolute difference in parsimony score induced by c on the two trees. In this note we prove that for binary trees there exists a character achieving this maximum that is convex on one of the trees (i.e. the parsimony score induced on that tree is equal to the number of states in the character minus 1) and such that the number of states in the character is at most 7d_MP - 5. This is the first non-trivial bound on the number of states required by optimal characters, convex or otherwise. The result potentially has algorithmic significance because, unlike general characters, convex characters with a bounded number of states can be enumerated in polynomial time.