On the Parameterized Complexity of the Perfect Phylogeny Problem

TL;DR

This paper investigates the parameterized complexity of the Perfect Phylogeny problem, establishing its completeness for the class XALP and providing complexity bounds under the Exponential Time Hypothesis.

Contribution

It classifies the parameterized complexity of Perfect Phylogeny and related problems as XALP-complete, introducing Triangulating Multicolored Graphs and proving tight complexity bounds.

Findings

Perfect Phylogeny is XALP-complete.

Triangulating Multicolored Graphs is XALP-complete.

No sub-exponential parameterized algorithms exist under ETH.

Abstract

This paper categorizes the parameterized complexity of the algorithmic problems Perfect Phylogeny and Triangulating Colored Graphs when parameterized by the number of genes and colors, respectively. We show that they are complete for the parameterized complexity class XALP using a reduction from Tree-chained Multicolor Independent Set and a proof of membership. We introduce the problem Triangulating Multicolored Graphs as a stepping stone and prove XALP-completeness for this problem as well. We also show that, assuming the Exponential Time Hypothesis, there exists no algorithm that solves any of these problems in time $f(k) n^{o(k)}$, where $n$ is the input size, $k$ the parameter, and $f$ any computable function.