A Practical Algorithm for Reconstructing Level-1 Phylogenetic Networks

Recently much attention has been devoted to the construction of phylogenetic networks which generalize phylogenetic trees in order to accommodate complex evolutionary processes. Here we present an efficient, practical algorithm for reconstructing level-1 phylogenetic networks - a type of network slightly more general than a phylogenetic tree - from triplets. Our algorithm has been made publicly available as the program LEV1ATHAN. It combines ideas from several known theoretical algorithms for phylogenetic tree and network reconstruction with two novel subroutines. Namely, an exponential-time exact and a greedy algorithm both of which are of independent theoretical interest. Most importantly, LEV1ATHAN runs in polynomial time and always constructs a level-1 network. If the data is consistent with a phylogenetic tree, then the algorithm constructs such a tree. Moreover, if the input triplet set is dense and, in addition, is fully consistent with some level-1 network, it will find such a network. The potential of LEV1ATHAN is explored by means of an extensive simulation study and a biological data set. One of our conclusions is that LEV1ATHAN is able to construct networks consistent with a high percentage of input triplets, even when these input triplets are affected by a low to moderate level of noise

Constructing level-2 phylogenetic networks from triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of species), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so to construct such a network. They also showed that, unlike in the case of trees (i.e. level-0 networks), the problem becomes NP-hard when the input is non-dense. Here we further extend this work by showing that, when the set of input triplets is dense, the problem is even polynomial-time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily non-tree like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also show that, in the non-dense case, the level-2 problem remains NP-hard

A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest

We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the Subtree Prune-and-Regraft (SPR) distance between two phylogenetic trees. Our result improves on the very recent 2.5-approximation algorithm due to Shi, Feng, You and Wang (2015). Our algorithm is the first approximation algorithm for this problem that uses LP duality in its analysis