Convex characters, algorithms and matchings

TL;DR

This paper advances algorithms in phylogenetics by integrating convex character enumeration with existing methods, leading to faster solutions for problems like maximum agreement forest and parsimony distance, and explores connections to matchings enumeration.

Contribution

It introduces improved algorithms for phylogenetic problems by combining convex character enumeration with parameterized algorithms and establishes links to matchings enumeration.

Findings

Faster exponential algorithms for maximum agreement forest problem.

Efficient computation of maximum parsimony distance using convex characters.

New results in enumeration of matchings on binary trees.

Abstract

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. In \cite{KelkS17} it was shown how to efficiently count, list and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterised algorithms can be used to speed up exponential-time algorithms for the \emph{maximum agreement forest problem} in phylogenetics. Second, we re-visit the quantity $g_2(T)$, defined as the number of convex characters on $T$ in which each state appears on at least 2 taxa. We use this to give an algorithm with running time $O( \phi^{n} \cdot \text{poly}(n) )$, where $\phi \approx 1.6181$ is the golden ratio and $n$ is the number of taxa in the input trees, for computation of \emph{maximum parsimony distance on two state characters}. By further restricting the characters counted by $g_2(T)$ we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to $O( 1.5895^{n} \cdot \text{poly}(n) )$, and obtain a number of new results in themselves relevant to enumeration of matchings on at-most binary trees.