A quadratic kernel for computing the hybridization number of multiple trees

It has recently been shown that the NP-hard problem of calculating the minimum number of hybridization events that is needed to explain a set of rooted binary phylogenetic trees by means of a hybridization network is fixed-parameter tractable if an instance of the problem consists of precisely two such trees. In this paper, we show that this problem remains fixed-parameter tractable for an arbitrarily large set of rooted binary phylogenetic trees. In particular, we present a quadratic kernel

A first step towards computing all hybridization networks for two rooted binary phylogenetic trees

Recently, considerable effort has been put into developing fast algorithms to reconstruct a rooted phylogenetic network that explains two rooted phylogenetic trees and has a minimum number of hybridization vertices. With the standard approach to tackle this problem being combinatorial, the reconstructed network is rarely unique. From a biological point of view, it is therefore of importance to not only compute one network, but all possible networks. In this paper, we make a first step towards approaching this goal by presenting the first algorithm---called allMAAFs---that calculates all maximum-acyclic-agreement forests for two rooted binary phylogenetic trees on the same set of taxa.Comment: 21 pages, 5 figure

Spaces of phylogenetic networks from generalized nearest-neighbor interchange operations

Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are used to represent the evolution of species which have undergone reticulate evolution. In this paper we consider spaces of such networks defined by some novel local operations that we introduce for converting one phylogenetic network into another. These operations are modeled on the well-studied nearest-neighbor interchange (NNI) operations on phylogenetic trees, and lead to natural generalizations of the tree spaces that have been previously associated to such operations. We present several results on spaces of some relatively simple networks, called level-1 networks, including the size of the neighborhood of a fixed network, and bounds on the diameter of the metric defined by taking the smallest number of operations required to convert one network into another.We expect that our results will be useful in the development of methods for systematically searching for optimal phylogenetic networks using, for example, likelihood and Bayesian approaches

A tight kernel for computing the tree bisection and reconnection distance between two phylogenetic trees

In 2001 Allen and Steel showed that, if subtree and chain reduction rules have been applied to two unrooted phylogenetic trees, the reduced trees will have at most 28k taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees. Here we reanalyse Allen and Steel's kernelization algorithm and prove that the reduced instances will in fact have at most 15k-9 taxa. Moreover we show, by describing a family of instances which have exactly 15k-9 taxa after reduction, that this new bound is tight. These instances also have no common clusters, showing that a third commonly-encountered reduction rule, the cluster reduction, cannot further reduce the size of the kernel in the worst case. To achieve these results we introduce and use "unrooted generators" which are analogues of rooted structures that have appeared earlier in the phylogenetic networks literature. Using similar argumentation we show that, for the minimum hybridization problem on two rooted trees, 9k-2 is a tight bound (when subtree and chain reduction rules have been applied) and 9k-4 is a tight bound (when, additionally, the cluster reduction has been applied) on the number of taxa, where k is the hybridization number of the two trees.Comment: One figure added, two small typos fixed. This version to appear in SIDMA (SIAM Journal on Discrete Mathematics