Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable

TL;DR

This paper proves that constructing minimal phylogenetic networks representing given clusters is fixed parameter tractable with respect to the reticulation number, extending previous polynomial results to more general networks.

Contribution

It introduces a fixed parameter tractability approach for constructing minimal phylogenetic networks based on reticulation number, generalizing prior polynomial-time results.

Findings

Determines existence of a level-<=k network in f(k)·poly(n) time

Extends fixed parameter tractability to networks with reticulation number k

Provides a constructive method for such networks

Abstract

Here we show that, given a set of clusters C on a set of taxa X, where |X|=n, it is possible to determine in time f(k).poly(n) whether there exists a level-<= k network (i.e. a network where each biconnected component has reticulation number at most k) that represents all the clusters in C in the softwired sense, and if so to construct such a network. This extends a polynomial time result from "On the elusiveness of clusters" by Kelk, Scornavacca and Van Iersel(2011). By generalizing the concept of "level-k generator" to general networks, we then extend this fixed parameter tractability result to the problem where k refers not to the level but to the reticulation number of the whole network.