arXiv subjects

Steven Kelk

Publications and source records attributed to Steven Kelk.

At least 37 records · Page 2Linked to original sources

arXiv2021

Convex characters, algorithms and matchings

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. In \cite{KelkS17} it was shown how to efficiently count, list and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterised algorithms can be used to speed up exponential-time algorithms for the \emph{maximum agreement forest problem} in phylogenetics. Second, we re-visit the quantity $g_2(T)$, defined as the number of convex characters on $T$ in which each state appears on at least 2 taxa. We use this to give an algorithm with running time $O( ϕ^{n} \cdot \text{poly}(n) )$, where $ϕ\approx 1.6181$ is the golden ratio and $n$ is the number of taxa in the input trees, for computation of \emph{maximum parsimony distance on two state characters}. By further restricting the characters counted by $g_2(T)$ we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to $O( 1.5895^{n} \cdot \text{poly}(n) )$, and obtain a number of new results in themselves relevant to enumeration of matchings on at-most binary trees.

math.CO↗

arXiv2021

Reflections on kernelizing and computing unrooted agreement forests

Phylogenetic trees are leaf-labelled trees used to model the evolution of species. Here we explore the practical impact of kernelization (i.e. data reduction) on the NP-hard problem of computing the TBR distance between two unrooted binary phylogenetic trees. This problem is better-known in the literature as the maximum agreement forest problem, where the goal is to partition the two trees into a minimum number of common, non-overlapping subtrees. We have implemented two well-known reduction rules, the subtree and chain reduction, and five more recent, theoretically stronger reduction rules, and compare the reduction achieved with and without the stronger rules. We find that the new rules yield smaller reduced instances and thus have clear practical added value. In many cases they also cause the TBR distance to decrease in a controlled fashion. Next, we compare the achieved reduction to the known worst-case theoretical bounds of 15k-9 and 11k-9 respectively, on the number of leaves of the two reduced trees, where k is the TBR distance, observing in both cases a far larger reduction in practice. As a by-product of our experimental framework we obtain a number of new insights into the actual computation of TBR distance. We find, for example, that very strong lower bounds on TBR distance can be obtained efficiently by randomly sampling certain carefully constructed partitions of the leaf labels, and identify instances which seem particularly challenging to solve exactly. The reduction rules have been implemented within our new solver Tubro which combines kernelization with an Integer Linear Programming (ILP) approach. Tubro also incorporates a number of additional features, such as a cluster reduction and a practical upper-bounding heuristic, and it can leverage combinatorial insights emerging from the proofs of correctness of the reduction rules to simplify the ILP.

cs.DS↗

arXiv2021

Sharp upper and lower bounds on a restricted class of convex characters

Let $\mathcal{T}$ be an unrooted binary tree with $n$ distinctly labelled leaves. Deriving its name from the field of phylogenetics, a convex character on $\mathcal{T}$ is simply a partition of the leaves such that the minimal spanning subtrees induced by the blocks of the partition are mutually disjoint. In earlier work Kelk and Stamoulis (Advances in Applied Mathematics 84 (2017), pp. 34--46) defined $g_k(\mathcal{T})$ as the number of convex characters where each block has at least $k$ leaves. Exact expressions were given for $g_1$ and $g_2$, where the topology of $\mathcal{T}$ turns out to be irrelevant, and it was noted that for $k \geq 3$ topological neutrality no longer holds. In this article, for every $k \geq 3$ we describe tree topologies achieving the maximum and minimum values of $g_k$ and determine corresponding expressions and exponential bounds for $g_k$. Finally, we reflect briefly on possible algorithmic applications of these results.

math.CO↗

arXiv2020

New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees

We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of $m$ binary trees with $n$ leaves each with a running time of $O(5^k\cdot n\cdot m)$, where $k$ is the minimum temporal hybridization number. We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most $d$ and at most $k$ reticulations in $O((8k)^d5^ k\cdot n\cdot m)$ time. Lastly, we introduce a $O(6^kk!\cdot k\cdot n^2)$ time algorithm for computing a minimum temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

cs.DS↗

arXiv2020

New reduction rules for the tree bisection and reconnection distance

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most 15k-9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here we propose five new reduction rules and show that these further reduce the bound to 11k-9. The new rules combine the ``unrooted generator'' approach introduced in [Kelk and Linz 2018] with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.

cs.DS↗

arXiv2020

Maximum parsimony distance on phylogenetictrees: a linear kernel and constant factor approximation algorithm

Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time constant-factor approximation algorithm; that the treewidth of a natural auxiliary graph structure encountered in phylogenetics is bounded by a function of the distance; and that the distance is within a constant factor of the size of a maximum agreement forest of the two trees, a well studied object in phylogenetics.

cs.DS↗

arXiv2019

Cutting an alignment with Ockham's razor

In this article, we investigate different parsimony-based approaches towards finding recombination breakpoints in a multiple sequence alignment. This recombination detection task is crucial in order to avoid errors in evolutionary analyses caused by mixing together portions of sequences which had a different evolution history. Following an overview of the field of recombination detection, we formulate four computational problems for this task with different objective functions. The four problems aim to minimize (1) the total homoplasy of all blocks (2) the maximum homoplasy per block (3) the total homoplasy ratio of all blocks and (4) the maximum homoplasy ratio per block. We describe algorithms for each of these problems, which are fixed-parameter tractable (FPT) when the characters are binary. We have implemented and tested the algorithms on simulated data, showing that minimizing the total homoplasy gives, in most cases, the most accurate results. Our implementation and experimental data have been made publicly available. Finally, we also consider the problem of combining blocks into non-contiguous blocks consisting of at most p contiguous parts. Fixing the homoplasy h of each block to 0, we show that this problem is NP-hard when p >= 3, but polynomial-time solvable for p = 2. Furthermore, the problem is FPT with parameter h for binary characters when p = 2. A number of interesting problems remain open.

q-bio.PE↗

arXiv2019

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.

cs.DS↗

arXiv2019

Foundations of Digital Archæoludology

Digital Archaeoludology (DAL) is a new field of study involving the analysis and reconstruction of ancient games from incomplete descriptions and archaeological evidence using modern computational techniques. The aim is to provide digital tools and methods to help game historians and other researchers better understand traditional games, their development throughout recorded human history, and their relationship to the development of human culture and mathematical knowledge. This work is being explored in the ERC-funded Digital Ludeme Project. The aim of this inaugural international research meeting on DAL is to gather together leading experts in relevant disciplines - computer science, artificial intelligence, machine learning, computational phylogenetics, mathematics, history, archaeology, anthropology, etc. - to discuss the key themes and establish the foundations for this new field of research, so that it may continue beyond the lifetime of its initiating project.

cs.AI↗

arXiv2019

Discovery of Important Subsequences in Electrocardiogram Beats Using the Nearest Neighbour Algorithm

The classification of time series data is a well-studied problem with numerous practical applications, such as medical diagnosis and speech recognition. A popular and effective approach is to classify new time series in the same way as their nearest neighbours, whereby proximity is defined using Dynamic Time Warping (DTW) distance, a measure analogous to sequence alignment in bioinformatics. However, practitioners are not only interested in accurate classification, they are also interested in why a time series is classified a certain way. To this end, we introduce here the problem of finding a minimum length subsequence of a time series, the removal of which changes the outcome of the classification under the nearest neighbour algorithm with DTW distance. Informally, such a subsequence is expected to be relevant for the classification and can be helpful for practitioners in interpreting the outcome. We describe a simple but optimized implementation for detecting these subsequences and define an accompanying measure to quantify the relevance of every time point in the time series for the classification. In tests on electrocardiogram data we show that the algorithm allows discovery of important subsequences and can be helpful in detecting abnormalities in cardiac rhythms distinguishing sick from healthy patients.

cs.LG↗

arXiv2019

Deciding the existence of a cherry-picking sequence is hard on two trees

Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability results for the minimum temporal-hybridization number problem. This is the optimization version of the problem where we wish to construct a temporal phylogenetic network that topologically embeds two given rooted binary phylogenetic trees and that has a minimum number of indegree-2 nodes, which represent events such as hybridization and horizontal gene transfer. We end on a positive note, pointing out that fixed parameter tractability results in this area are likely to ensure the continued relevance of the temporal phylogenetic network model.

q-bio.PE↗

arXiv2019

A third strike against perfect phylogeny

Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam's Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result states that 2-state characters permit a perfect phylogeny precisely if each subset of 2 characters permits one. More recently, it was shown that for 3-state characters the same property holds but for size-3 subsets. A long-standing open problem asked whether such a constant exists for each number of states. More precisely, it has been conjectured that for any fixed integer $r$, there exists a constant $f(r)$ such that a set of $r$-state characters $C$ has a perfect phylogeny if and only if every subset of at most $f(r)$ characters has a perfect phylogeny. In this paper, we show that this conjecture is false. In particular, we show that for any constant $t$, there exists a set $C$ of $8$-state characters such that $C$ has no perfect phylogeny, but there exists a perfect phylogeny for every subset of $t$ characters. This negative result complements the two negative results ("strikes") of Bodlaender et al. We reflect on the consequences of this third strike, pointing out that while it does close off some routes for efficient algorithm development, many others remain open.

cs.DM↗

arXiv2019

Integrality Gaps for Colorful Matchings

We study the integrality gap of the natural linear programming relaxation for the \textit{Bounded Color Matching} (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Füredi [\emph{Combinatorica}, 1(2):155-162, 1981]. We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem.

cs.DM↗

arXiv2018

Treewidth of display graphs: bounds, brambles and applications

Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics.

cs.DS↗

arXiv2017

Finding the most parsimonious or likely tree in a network with respect to an alignment

Phylogenetic networks are often constructed by merging multiple conflicting phylogenetic signals into a directed acyclic graph. It is interesting to explore whether a network constructed in this way induces biologically-relevant phylogenetic signals that were not present in the input. Here we show that, given a multiple alignment A for a set of taxa X and a rooted phylogenetic network N whose leaves are labelled by X, it is NP-hard to locate the most parsimonious phylogenetic tree displayed by N (with respect to A) even when the level of N - the maximum number of reticulation nodes within a biconnected component - is 1 and A contains only 2 distinct states. (If, additionally, gaps are allowed the problem becomes APX-hard.) We also show that under the same conditions, and assuming a simple binary symmetric model of character evolution, finding the most likely tree displayed by the network is NP-hard. These negative results contrast with earlier work on parsimony in which it is shown that if A consists of a single column the problem is fixed parameter tractable in the level. We conclude with a discussion of why, despite the NP-hardness, both the parsimony and likelihood problem can likely be well-solved in practice.

q-bio.PE↗

arXiv2017

Treewidth distance on phylogenetic trees

In this article we study the treewidth of the \emph{display graph}, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules which are commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms. In some cases (the \emph{subtree} reduction) the reduction rules behave similarly with respect to treewidth, while others (the \emph{cluster} reduction) behave very differently, and the behaviour of the \emph{chain reduction} is particularly intriguing because of its link with graph separators and forbidden minors. We also show that the gap between treewidth and Tree Bisection and Reconnect (TBR) distance can be infinitely large, and that unlike, for example, planar graphs the treewidth of the display graph can be as much as linear in its number of vertices. On a slightly different note we show that if a display graph is formed from the fusion of a phylogenetic network and a tree, rather than from two trees, the treewidth of the display graph is bounded whenever the tree can be topologically embedded ("displayed") within the network. This opens the door to the formulation of the display problem in Monadic Second Order Logic (MSOL). A number of other auxiliary results are given. We conclude with a discussion and list a number of open problems.

cs.DM↗

arXiv2016

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.

cs.DS↗

arXiv2016

A note on convex characters, Fibonacci numbers and exponential-time algorithms

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. Given an unrooted, binary phylogenetic tree T on a set of n >= 2 taxa, a closed (but fairly opaque) expression for the number of convex characters on T has been known since 1992, and this is independent of the exact topology of T. In this note we prove that this number is actually equal to the (2n-1)th Fibonacci number. Next, we define g_k(T) to be the number of convex characters on T in which each state appears on at least k taxa. We show that, somewhat curiously, g_2(T) is also independent of the topology of T, and is equal to to the (n-1)th Fibonacci number. As we demonstrate, this topological neutrality subsequently breaks down for k >= 3. However, we show that for each fixed k >= 1, g_k(T) can be computed in O(n) time and the set of characters thus counted can be efficiently listed and sampled. We use these insights to give a simple but effective exact algorithm for the NP-hard maximum parsimony distance problem that runs in time $Θ( ϕ^{n} \cdot n^2 )$, where $ϕ\approx 1.618...$ is the golden ratio, and an exact algorithm which computes the tree bisection and reconnection distance (equivalently, a maximum agreement forest) in time $Θ( ϕ^{2n}\cdot \text{poly}(n))$, where $ϕ^2 \approx 2.619$.

q-bio.PE↗