On unrooted and root-uncertain variants of several well-known phylogenetic network problems

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an \emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted trees. In the third part of the paper we consider the "root uncertain" variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure

A Fast Quartet Tree Heuristic for Hierarchical Clustering

The Minimum Quartet Tree Cost problem is to construct an optimal weight tree from the $3{n \choose 4}$ weighted quartet topologies on $n$ objects, where optimality means that the summed weight of the embedded quartet topologies is optimal (so it can be the case that the optimal tree embeds all quartets as nonoptimal topologies). We present a Monte Carlo heuristic, based on randomized hill climbing, for approximating the optimal weight tree, given the quartet topology weights. The method repeatedly transforms a dendrogram, with all objects involved as leaves, achieving a monotonic approximation to the exact single globally optimal tree. The problem and the solution heuristic has been extensively used for general hierarchical clustering of nontree-like (non-phylogeny) data in various domains and across domains with heterogeneous data. We also present a greatly improved heuristic, reducing the running time by a factor of order a thousand to ten thousand. All this is implemented and available, as part of the CompLearn package. We compare performance and running time of the original and improved versions with those of UPGMA, BioNJ, and NJ, as implemented in the SplitsTree package on genomic data for which the latter are optimized. Keywords: Data and knowledge visualization, Pattern matching--Clustering--Algorithms/Similarity measures, Hierarchical clustering, Global optimization, Quartet tree, Randomized hill-climbing,Comment: LaTeX, 40 pages, 11 figures; this paper has substantial overlap with arXiv:cs/0606048 in cs.D

Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots

Although taxonomy is often used informally to evaluate the results of phylogenetic inference and find the root of phylogenetic trees, algorithmic methods to do so are lacking. In this paper we formalize these procedures and develop algorithms to solve the relevant problems. In particular, we introduce a new algorithm that solves a "subcoloring" problem for expressing the difference between the taxonomy and phylogeny at a given rank. This algorithm improves upon the current best algorithm in terms of asymptotic complexity for the parameter regime of interest; we also describe a branch-and-bound algorithm that saves orders of magnitude in computation on real data sets. We also develop a formalism and an algorithm for rooting phylogenetic trees according to a taxonomy. All of these algorithms are implemented in freely-available software.Comment: Version submitted to Algorithms for Molecular Biology. A number of fixes from previous versio

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees

The relationship between two important problems in tree pattern matching, the largest common subtree and the smallest common supertree problems, is established by means of simple constructions, which allow one to obtain a largest common subtree of two trees from a smallest common supertree of them, and vice versa. These constructions are the same for isomorphic, homeomorphic, topological, and minor embeddings, they take only time linear in the size of the trees, and they turn out to have a clear algebraic meaning.Comment: 32 page

Enumerating All Maximal Frequent Subtrees

Given a collection of leaf-labeled trees on a common leafset and a fraction f in (1/2,1], a frequent subtree (FST) is a subtree isomorphically included in at least fraction f of the input trees. The well-known maximum agreement subtree (MAST) problem identifies FST with f = 1 and having the largest number of leaves. Apart from its intrinsic interest from the algorithmic perspective, MAST has practical applications as a metric for tree similarity, for computing tree congruence, in detection horizontal gene transfer events and as a consensus approach. Enumerating FSTs extend the MAST problem by denition and reveal additional subtrees not displayed by MAST. This can happen in tow ways - such a subtree is included in majority but not all of the input trees or such a subtree though included in all the input trees, does not have the maximum number of leaves. Further, FSTs can be enumerated on collection o ftrees having partially overlapping leafsets. MAST may not be useful here especially if the common overlap among leafsets is very low. Though very useful, the number of FSTs suffer from combinatorial explosion - just a single enumeration of maximal frequent subtrees (MFSTs). A MFST is a FST that is not a subtree to any othe rFST. the set of MFSTs is a compact non-redundant summary of all FSTs and is much smaller in size. Here we tackle the novel problem of enumerating all MFSTs in collections of phylogenetic trees. We demonstrate its utility in returning larger consensus trees in comparison to MAST. The current implementation is available on the web