Finding the most parsimonious or likely tree in a network with respect to an alignment

TL;DR

This paper proves that finding the most parsimonious or likely phylogenetic tree within a network, given an alignment, is NP-hard even under simplified conditions, highlighting computational challenges in phylogenetic analysis.

Contribution

It establishes NP-hardness of locating optimal trees in phylogenetic networks with respect to alignments, even at level 1 and with minimal states, contrasting previous tractability results.

Findings

NP-hardness of parsimony and likelihood tree selection in networks

Complexity persists even with simplified network and data conditions

Discussion on practical solvability despite theoretical hardness

Abstract

Phylogenetic networks are often constructed by merging multiple conflicting phylogenetic signals into a directed acyclic graph. It is interesting to explore whether a network constructed in this way induces biologically-relevant phylogenetic signals that were not present in the input. Here we show that, given a multiple alignment A for a set of taxa X and a rooted phylogenetic network N whose leaves are labelled by X, it is NP-hard to locate the most parsimonious phylogenetic tree displayed by N (with respect to A) even when the level of N - the maximum number of reticulation nodes within a biconnected component - is 1 and A contains only 2 distinct states. (If, additionally, gaps are allowed the problem becomes APX-hard.) We also show that under the same conditions, and assuming a simple binary symmetric model of character evolution, finding the most likely tree displayed by the network is NP-hard. These negative results contrast with earlier work on parsimony in which it is shown that if A consists of a single column the problem is fixed parameter tractable in the level. We conclude with a discussion of why, despite the NP-hardness, both the parsimony and likelihood problem can likely be well-solved in practice.