Reconstructing a phylogenetic level-1 network from quartets

TL;DR

This paper introduces algorithms to reconstruct unrooted binary phylogenetic level-1 networks from quartet data efficiently, using linear algebra over GF(2), with proven complexity bounds and consistency guarantees.

Contribution

It presents two algorithms for reconstructing level-1 networks from quartets, including a more general method handling diverse data and providing complexity and correctness guarantees.

Findings

O(n^3) algorithm for fixed-taxon quartets

O(n^6) algorithm for general quartet data

Algorithm guarantees consistency or provides a certificate of inconsistency

Abstract

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on n taxa from the set of all quartets containing a certain fixed taxon, in O(n^3) time. We also present a more general method which can handle more diverse quartet data, but which takes O(n^6) time. Both methods proceed by solving a certain system of linear equations over GF(2). For a general dense quartet set (containing at least one quartet on every four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an (O(n^2) sized) certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set.