arXiv subjects

Steven Kelk

Publications and source records attributed to Steven Kelk.

At least 55 records · Page 3Linked to original sources

arXiv2016

Do branch lengths help to locate a tree in a phylogenetic network?

Phylogenetic networks are increasingly used in evolutionary biology to represent the history of species that have undergone reticulate events such as horizontal gene transfer, hybrid speciation and recombination. One of the most fundamental questions that arise in this context is whether the evolution of a gene with one copy in all species can be explained by a given network. In mathematical terms, this is often translated in the following way: is a given phylogenetic tree contained in a given phylogenetic network? Recently this tree containment problem has been widely investigated from a computational perspective, but most studies have only focused on the topology of the phylo- genies, ignoring a piece of information that, in the case of phylogenetic trees, is routinely inferred by evolutionary analyses: branch lengths. These measure the amount of change (e.g., nucleotide substitutions) that has occurred along each branch of the phylogeny. Here, we study a number of versions of the tree containment problem that explicitly account for branch lengths. We show that, although length information has the potential to locate more precisely a tree within a network, the problem is computationally hard in its most general form. On a positive note, for a number of special cases of biological relevance, we provide algorithms that solve this problem efficiently. This includes the case of networks of limited complexity, for which it is possible to recover, among the trees contained by the network with the same topology as the input tree, the closest one in terms of branch lengths.

q-bio.PE↗

arXiv2016

Reduction rules for the maximum parsimony distance on phylogenetic trees

In phylogenetics, distances are often used to measure the incongruence between a pair of phylogenetic trees that are reconstructed by different methods or using different regions of genome. Motivated by the maximum parsimony principle in tree inference, we recently introduced the maximum parsimony (MP) distance, which enjoys various attractive properties due to its connection with several other well-known tree distances, such as TBR and SPR. Here we show that computing the MP distance between two trees, a NP-hard problem in general, is fixed parameter tractable in terms of the TBR distance between the tree pair. Our approach is based on two reduction rules--the chain reduction and the subtree reduction--that are widely used in computing TBR and SPR distances. More precisely, we show that reducing chains to length 4 (but not shorter) preserves the MP distance. In addition, we describe a generalization of the subtree reduction which allows the pendant subtrees to be rooted in different places, and show that this still preserves the MP distance. On a slightly different note we also show that Monadic Second Order Logic (MSOL), posited over an auxiliary graph structure known as the display graph (obtained by merging the two trees at their leaves), can be used to obtain an alternative proof that computation of MP distance is fixed parameter tractable in terms of TBR-distance. We conclude with an extended discussion in which we focus on similarities and differences between MP distance and TBR distance and present a number of open problems. One particularly intriguing question, emerging from the MSOL formulation, is whether two trees with bounded MP distance induce display graphs of bounded treewidth.

q-bio.PE↗

arXiv2016

Hybridization Number on Three Rooted Binary Trees is EPT

Phylogenetic networks are leaf-labelled directed acyclic graphs that are used to describe non-treelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all indegrees minus the number of nodes plus one. The Hybridization Number problem takes as input a collection of phylogenetic trees and asks to construct a phylogenetic network that contains an embedding of each of the input trees and has a smallest possible hybridization number. We present an algorithm for the Hybridization Number problem on three binary trees on $n$ leaves, which runs in time $O(c^k poly(n))$, with $k$ the hybridization number of an optimal network and $c$ a constant. For two trees, an algorithm with running time $O(3.18^k n)$ was proposed before whereas an algorithm with running time $O(c^k poly(n))$ had prior to this article remained elusive for more than two trees. The algorithm for two trees uses the close connection to acyclic agreement forests to achieve a linear exponent in the running time, while previous algorithms for more than two trees (explicitly or implicitly) relied on a brute force search through all possible underlying network topologies, leading to running times that are not $O(c^k poly(n))$ for any $c$. The connection to acyclic agreement forests is much weaker for more than two trees, so even given the right agreement forest, reconstructing the network poses major challenges. We prove novel structural results that allow us to reconstruct a network without having to guess the underlying topology. Our techniques generalize to more than three input trees with the exception of one key lemma that maps nodes in the network to tree nodes and, thus, minimizes the amount of guessing involved in constructing the network. The main open problem therefore is to establish a similar mapping for more than three trees.

cs.DS↗

arXiv2016

Kernelizations for the hybridization number problem on multiple nonbinary trees

Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic trees on $X$ and an integer $k$, the Hybridization Number problem asks if there exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(Δ^+-1)$ respectively, with $t$ the number of input trees and $Δ^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$.

cs.DM↗

arXiv2015

A linear bound on the number of states in optimal convex characters for maximum parsimony distance

Given two phylogenetic trees on the same set of taxa X, the maximum parsimony distance d_MP is defined as the maximum, ranging over all characters c on X, of the absolute difference in parsimony score induced by c on the two trees. In this note we prove that for binary trees there exists a character achieving this maximum that is convex on one of the trees (i.e. the parsimony score induced on that tree is equal to the number of states in the character minus 1) and such that the number of states in the character is at most 7d_MP - 5. This is the first non-trivial bound on the number of states required by optimal characters, convex or otherwise. The result potentially has algorithmic significance because, unlike general characters, convex characters with a bounded number of states can be enumerated in polynomial time.

q-bio.PE↗

arXiv2015

Phylogenetic incongruence through the lens of Monadic Second Order logic

Within the field of phylogenetics there is growing interest in measures for summarising the dissimilarity, or 'incongruence', of two or more phylogenetic trees. Many of these measures are NP-hard to compute and this has stimulated a considerable volume of research into fixed parameter tractable algorithms. In this article we use Monadic Second Order logic (MSOL) to give alternative, compact proofs of fixed parameter tractability for several well-known incongruency measures. In doing so we wish to demonstrate the considerable potential of MSOL - machinery still largely unknown outside the algorithmic graph theory community - within phylogenetics. A crucial component of this work is the observation that many of these measures, when bounded, imply the existence of an 'agreement forest' of bounded size, which in turn implies that an auxiliary graph structure, the display graph, has bounded treewidth. It is this bound on treewidth that makes the machinery of MSOL available for proving fixed parameter tractability. We give a variety of different MSOL formulations. Some are based on explicitly encoding agreement forests, while some only use them implicitly to generate the treewidth bound. Our formulations introduce a number of "phylogenetics MSOL primitives" which will hopefully be of use to other researchers.

cs.DS↗

arXiv2015

On the complexity of computing MP distance between binary phylogenetic trees

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two \emph{binary} phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic trees, and it is particularly relevant given the prominence of binary trees in the phylogenetics literature. As a corollary to the main hardness result we show that computation of MP distance is also hard on binary trees if the number of states available is bounded. In fact, via a different reduction we show that it is hard even if only two states are available. Finally, as a first response to this hardness we give a simple Integer Linear Program (ILP) formulation which is capable of computing the MP distance exactly for small trees (and for larger trees when only a small number of character states are available) and which is used to computationally verify several auxiliary results required by the hardness proofs.

q-bio.PE↗

arXiv2014

Exact reconciliation of undated trees

Reconciliation methods aim at recovering macro evolutionary events and at localizing them in the species history, by observing discrepancies between gene family trees and species trees. In this article we introduce an Integer Linear Programming (ILP) approach for the NP-hard problem of computing a most parsimonious time-consistent reconciliation of a gene tree with a species tree when dating information on speciations is not available. The ILP formulation, which builds upon the DTL model, returns a most parsimonious reconciliation ranging over all possible datings of the nodes of the species tree. By studying its performance on plausible simulated data we conclude that the ILP approach is significantly faster than a brute force search through the space of all possible species tree datings. Although the ILP formulation is currently limited to small trees, we believe that it is an important proof-of-concept which opens the door to the possibility of developing an exact, parsimony based approach to dating species trees. The software (ILPEACE) is freely available for download.

q-bio.PE↗

arXiv2014

Satisfying ternary permutation constraints by multiple linear orders or phylogenetic trees

A ternary permutation constraint satisfaction problem (CSP) is specified by a subset Pi of the symmetric group S_3. An instance of such a problem consists of a set of variables V and a set of constraints C, where each constraint is an ordered triple of distinct elements from V. The goal is to construct a linear order alpha on V such that, for each constraint (a,b,c) in C, the ordering of a,b,c induced by alpha is in Pi. Excluding symmetries and trivial cases there are 11 such problems, and their complexity is well known. Here we consider the variant of the problem, denoted 2-Pi, where we are allowed to construct two linear orders alpha and beta and each constraint needs to be satisfied by at least one of the two. We give a full complexity classification of all 11 2-Pi problems, observing that in the switch from one to two linear orders the complexity landscape changes quite abruptly and that hardness proofs become rather intricate. We then focus on one of the 11 problems in particular, which is closely related to the '2-Caterpillar Compatibility' problem in the phylogenetics literature. We show that this particular CSP remains hard on three linear orders, and also in the biologically relevant case when we swap three linear orders for three phylogenetic trees, yielding the '3-Tree Compatibility' problem. Due to the biological relevance of this problem we also give extremal results concerning the minimum number of trees required, in the worst case, to satisfy a set of rooted triplet constraints on n leaf labels.

cs.CC↗

arXiv2014

A practical approximation algorithm for solving massive instances of hybridization number for binary and nonbinary trees

Reticulate events play an important role in determining evolutionary relationships. The problem of computing the minimum number of such events to explain discordance between two phylogenetic trees is a hard computational problem. Even for binary trees, exact solvers struggle to solve instances with reticulation number larger than 40-50. Here we present CycleKiller and NonbinaryCycleKiller, the first methods to produce solutions verifiably close to optimality for instances with hundreds or even thousands of reticulations. Using simulations, we demonstrate that these algorithms run quickly for large and difficult instances, producing solutions that are very close to optimality. As a spin-off from our simulations we also present TerminusEst, which is the fastest exact method currently available that can handle nonbinary trees: this is used to measure the accuracy of the NonbinaryCycleKiller algorithm. All three methods are based on extensions of previous theoretical work and are publicly available. We also apply our methods to real data.

q-bio.PE↗

arXiv2014

On Computing the Maximum Parsimony Score of a Phylogenetic Network

Phylogenetic networks are used to display the relationship of different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as Maximum Parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of Maximum Parsimony on networks, "hardwired" and "softwired", and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multicut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and softwired parsimony score can be computed efficiently using Integer Linear Programming. The software has been made freely available.

q-bio.PE↗

arXiv2014

On low treewidth graphs and supertrees

Compatibility of unrooted phylogenetic trees is a well studied problem in phylogenetics. It asks to determine whether for a set of k input trees there exists a larger tree (called a supertree) that contains the topologies of all k input trees. When any such supertree exists we call the instance compatible and otherwise incompatible. It is known that the problem is NP-hard and FPT, although a constructive FPT algorithm is not known. It has been shown that whenever the treewidth of an auxiliary structure known as the display graph is strictly larger than the number of input trees, the instance is incompatible. Here we show that whenever the treewidth of the display graph is at most 2, the instance is compatible. Furthermore, we give a polynomial-time algorithm to construct a supertree in this case. Finally, we demonstrate both compatible and incompatible instances that have display graphs with treewidth 3, highlighting that the treewidth of the display graph is (on its own) not sufficient to determine compatibility.

cs.DM↗

arXiv2014

On the Maximum Parsimony distance between phylogenetic trees

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Here, based on an idea of Bruen and Bryant, we propose and analyze a new distance measure: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. In this article we show that this new distance is a metric and provides a lower bound to the well-known Subtree Prune and Regraft (SPR) distance. We also show that to compute the MP distance it is sufficient to consider only characters that are convex on one of the trees, and prove several additional structural properties of the distance. On the complexity side, we prove that calculating the MP distance is in general NP-hard, and identify an interesting island of tractability in which the distance can be calculated in polynomial time.

q-bio.PE↗

arXiv2013

A short note on exponential-time algorithms for hybridization number

In this short note we prove that, given two (not necessarily binary) rooted phylogenetic trees T_1, T_2 on the same set of taxa X, where |X|=n, the hybridization number of T_1 and T_2 can be computed in time O^{*}(2^n) i.e. O(2^{n} poly(n)). The result also means that a Maximum Acyclic Agreement Forest (MAAF) can be computed within the same time bound.

q-bio.PE↗

arXiv2013

Fighting network space: it is time for an SQL-type language to filter phylogenetic networks

The search space of rooted phylogenetic trees is vast and a major research focus of recent decades has been the development of algorithms to effectively navigate this space. However this space is tiny when compared with the space of rooted phylogenetic networks, and navigating this enlarged space remains a poorly understood problem. This, and the difficulty of biologically interpreting such networks, obstructs adoption of networks as tools for modelling reticulation. Here, we argue that the superimposition of biologically motivated constraints, via an SQL-style language, can both stimulate use of network software by biologists and potentially significantly prune the search space.

q-bio.PE↗

arXiv2012

Approximation algorithms for nonbinary agreement forests

Given two rooted phylogenetic trees on the same set of taxa X, the Maximum Agreement Forest problem (MAF) asks to find a forest that is, in a certain sense, common to both trees and has a minimum number of components. The Maximum Acyclic Agreement Forest problem (MAAF) has the additional restriction that the components of the forest cannot have conflicting ancestral relations in the input trees. There has been considerable interest in the special cases of these problems in which the input trees are required to be binary. However, in practice, phylogenetic trees are rarely binary, due to uncertainty about the precise order of speciation events. Here, we show that the general, nonbinary version of MAF has a polynomial-time 4-approximation and a fixed-parameter tractable (exact) algorithm that runs in O(4^k poly(n)) time, where n = |X| and k is the number of components of the agreement forest minus one. Moreover, we show that a c-approximation algorithm for nonbinary MAF and a d-approximation algorithm for the classical problem Directed Feedback Vertex Set (DFVS) can be combined to yield a d(c+3)-approximation for nonbinary MAAF. The algorithms for MAF have been implemented and made publicly available.

math.CO↗

arXiv2012

Towards the fixed parameter tractability of constructing minimal phylogenetic networks from arbitrary sets of nonbinary trees

It has remained an open question for some time whether, given a set of not necessarily binary (i.e. "nonbinary") trees T on a set of taxa X, it is possible to determine in time f(r).poly(m) whether there exists a phylogenetic network that displays all the trees in T, where r refers to the reticulation number of the network and m=|X|+|T|. Here we show that this holds if one or both of the following conditions holds: (1) |T| is bounded by a function of r; (2) the maximum degree of the nodes in T is bounded by a function of r. These sufficient conditions absorb and significantly extend known special cases, namely when all the trees in T are binary, or T contains exactly two nonbinary trees. We believe this result is an important step towards settling the issue for an arbitrarily large and complex set of nonbinary trees. For completeness we show that the problem is certainly solveable in polynomial time.

q-bio.PE↗

arXiv2012

A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees

Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted, not necessarily binary phylogenetic trees on taxon set X in time (6^r.r!).poly(n)$, where n=|X|. The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on X, and several insights from the softwired clusters literature. This yields a surprisingly simple and practical bounded-search algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem.

q-bio.QM↗