A Near-Linear Kernel for Two-Parsimony Distance

TL;DR

This paper proves that computing the bounded-state maximum parsimony distance between two phylogenetic trees is fixed-parameter tractable with a near-linear kernel size, extending previous results for the unbounded case.

Contribution

The paper establishes a fixed-parameter tractable algorithm with an $O(k \, ext{lg} \, k)$ kernel size for computing the bounded-state maximum parsimony distance between phylogenetic trees, generalizing prior work.

Findings

Kernel size for the problem is $O(k \, ext{lg} \, k)$.

Introduces the concept of leg-disjoint incompatible quartets.

Proves fixed-parameter tractability for all $t$.

Abstract

The maximum parsimony distance $d_{\textrm{MP}}(T_1,T_2)$ and the bounded-state maximum parsimony distance $d_{\textrm{MP}}^t(T_1,T_2)$ measure the difference between two phylogenetic trees $T_1,T_2$ in terms of the maximum difference between their parsimony scores for any character (with $t$ a bound on the number of states in the character, in the case of $d_{\textrm{MP}}^t(T_1,T_2)$). While computing $d_{\textrm{MP}}(T_1, T_2)$ was previously shown to be fixed-parameter tractable with a linear kernel, no such result was known for $d_{\textrm{MP}}^t(T_1,T_2)$. In this paper, we prove that computing $d_{\textrm{MP}}^t(T_1, T_2)$ is fixed-parameter tractable for all~$t$. Specifically, we prove that this problem has a kernel of size $O(k \lg k)$, where $k = d_{\textrm{MP}}^t(T_1, T_2)$. As the primary analysis tool, we introduce the concept of leg-disjoint incompatible quartets, which may be of independent interest.