Kernelizations for the hybridization number problem on multiple nonbinary trees

TL;DR

This paper introduces kernelization algorithms for the Hybridization Number problem on multiple nonbinary trees, providing theoretical bounds and practical algorithms, along with an efficient fixed-parameter algorithm.

Contribution

The paper presents two new kernelization algorithms with size bounds and an $n^{f(k)}t$-time algorithm for the Hybridization Number problem on multiple nonbinary trees.

Findings

Kernelization algorithms with sizes $4k(5k)^t$ and $20k^2( ext{max outdegree})$.

Practical relevance demonstrated through experiments on simulated data.

An $n^{f(k)}t$-time algorithm for the problem.

Abstract

Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic trees on $X$ and an integer $k$, the Hybridization Number problem asks if there exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$ respectively, with $t$ the number of input trees and $\Delta^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$.