Optimizing Phylogenetic Supertrees Using Answer Set Programming

The supertree construction problem is about combining several phylogenetic trees with possibly conflicting information into a single tree that has all the leaves of the source trees as its leaves and the relationships between the leaves are as consistent with the source trees as possible. This leads to an optimization problem that is computationally challenging and typically heuristic methods, such as matrix representation with parsimony (MRP), are used. In this paper we consider the use of answer set programming to solve the supertree construction problem in terms of two alternative encodings. The first is based on an existing encoding of trees using substructures known as quartets, while the other novel encoding captures the relationships present in trees through direct projections. We use these encodings to compute a genus-level supertree for the family of cats (Felidae). Furthermore, we compare our results to recent supertrees obtained by the MRP method.Comment: To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of ICLP 201

Constraints, Lazy Constraints, or Propagators in ASP Solving: An Empirical Analysis

Answer Set Programming (ASP) is a well-established declarative paradigm. One of the successes of ASP is the availability of efficient systems. State-of-the-art systems are based on the ground+solve approach. In some applications this approach is infeasible because the grounding of one or few constraints is expensive. In this paper, we systematically compare alternative strategies to avoid the instantiation of problematic constraints, that are based on custom extensions of the solver. Results on real and synthetic benchmarks highlight some strengths and weaknesses of the different strategies. (Under consideration for acceptance in TPLP, ICLP 2017 Special Issue.)Comment: Paper presented at the 33nd International Conference on Logic Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017. 16 page

Building a Small and Informative Phylogenetic Supertree

We combine two fundamental, previously studied optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency (MAXRTC) and minimally resolved supertree (MINRS) into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). The input to our new problem is a set R of resolved triplets (rooted, binary phylogenetic trees with three leaves each) and the objective is to find a phylogenetic tree with exactly q internal nodes that contains the largest possible number of triplets from R. We first prove that q-MAXRTC is NP-hard even to approximate within a constant ratio for every fixed q >= 2, and then develop various polynomial-time approximation algorithms for different values of q. Next, we show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much triplet branching information. As an extreme example, we show that allowing only nine internal nodes is still sufficient to capture on average 80% of the rooted triplets from some recently published trees, each having between 760 and 3081 internal nodes. Finally, to demonstrate the algorithmic advantage of using trees with few internal nodes, we propose a new algorithm for computing the rooted triplet distance between two phylogenetic trees over a leaf label set of size n that runs in O(q n) time, where q is the number of internal nodes in the smaller tree, and is therefore faster than the currently best algorithms for the problem (with O(n log n) time complexity [SODA 2013, ESA 2017]) whenever q = o(log n)

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

Fast and accurate supertrees: towards large scale phylogenies

Phylogenetics is the study of evolutionary relationships between biological entities; phylogenetic trees (phylogenies) are a visualization of these evolutionary relationships. Accurate approaches to reconstruct hylogenies from sequence data usually result in NPhard optimization problems, hence local search heuristics have to be applied in practice. These methods are highly accurate and fast enough as long as the input data is not too large. Divide-and-conquer techniques are a promising approach to boost scalability and accuracy of those local search heuristics on very large datasets. A divide-and-conquer method breaks down a large phylogenetic problem into smaller sub-problems that are computationally easier to solve. The sub-problems (overlapping trees) are then combined using a supertree method. Supertree methods merge a set of overlapping phylogenetic trees into a supertree containing all taxa of the input trees. The challenge in supertree reconstruction is the way of dealing with conflicting information in the input trees. Many different algorithms for different objective functions have been suggested to resolve these conflicts. In particular, there are methods that encode the source trees in a matrix and the supertree is constructed applying a local search heuristic to optimize the respective objective function. The most widely used supertree methods use such local search heuristics. However, to really improve the scalability of accurate tree reconstruction by divide-and-conquer approaches, accurate polynomial time methods are needed for the supertree reconstruction step. In this work, we present approaches for accurate polynomial time supertree reconstruction in particular Bad Clade Deletion (BCD), a novel heuristic supertree algorithm with polynomial running time. BCD uses minimum cuts to greedily delete a locally minimal number of columns from a matrix representation to make it compatible. Different from local search heuristics, it guarantees to return the directed perfect phylogeny for the input matrix, corresponding to the parent tree of the input trees if one exists. BCD can take support values of the source trees into account without an increase in complexity. We show how reliable clades can be used to restrict the search space for BCD and how those clades can be collected from the input data using the Greedy Strict Consensus Merger. Finally, we introduce a beam search extension for the BCD algorithm that keeps alive a constant number of partial solutions in each top-down iteration phase. The guaranteed worst-case running time of BCD with beam search extension is still polynomial. We present an exact and a randomized subroutine to generate suboptimal partial solutions. In our thorough evaluation on several simulated and biological datasets against a representative set of supertree methods we found that BCD is more accurate than the most accurate supertree methods when using support values and search space restriction on simulated data. Simultaneously BCD is faster than any other evaluated method. The beam search approach improved the accuracy of BCD on all evaluated datasets at the cost of speed. We found that BCD supertrees can boost maximum likelihood tree reconstruction when used as starting tree. Further, BCD could handle large scale datasets where local search heuristics did not converge in reasonable time. Due to its combination of speed, accuracy, and the ability to reconstruct the parent tree if one exists, BCD is a promising approach to enable outstanding scalability of divide-and-conquer approaches.Die Phylogenetik studiert die evolutionären Beziehungen zwischen biologischen Entitäten. Phylogenetische Bäume sind eine Visualisierung dieser Beziehungen. Akkurate Ansätze zur Rekonstruktion von Phylogenien aus Sequenzdaten führen in der Regel zu NP-schweren Optimierungsproblemen, sodass in der Praxis lokale Suchheuristiken angewendet werden müssen. Diese Methoden liefern akkurate Bäume und sind schnell genug, solange die Eingabedaten nicht zu groß werden. Teile-und-herrsche-Verfahren sind ein vielversprechender Ansatz, um Skalierbarkeit und Genauigkeit dieser lokalen Suchheuristiken auf sehr großen Datensätzen zu verbessern. Beim Teile-und-herrsche-Ansatz zerlegt man ein großes phylogenetisches Problem in kleinere Teilprobleme, die einfacher und schneller zu lösen sind. Die Teilprobleme, in diesem Fall überlappende Teilbäume, müssen dann zu einem gesamtheitlichen Baum kombiniert werden. Superbaummethoden verschmelzen solche überlappenden phylogenetischen Bäume zu einem Superbaum, der alle Taxa der Eingangsbäume enthält. Die Herausforderung bei der Superbaumrekonstruktion besteht darin, mit widersprüchlichen Eingabebäumen umzugehen. Es wurden viele verschiedene Algorithmen mit unterschiedlichen Zielfunktionen entwickelt, um solche Widersprüche möglichst sinnvoll aufzulösen. Verfahren, die auf der Kodierung der Eingabebäume als Matrixrepräsentation basieren, sind am weitesten verbreitet. Die zum Auflösen der Konflikte verwendeten Zielfunktionen führen in der Regel zu NP-schweren Optimierungsproblemen, sodass in der Praxis auch hier lokale Suchheuristiken zum Einsatz kommen. Da diese Ansätze nicht wesentlich besser mit der Größe der Eingabedaten skalieren als die direkte Rekonstruktion aus Sequenzdaten, werden für die Superbaumrekonstruktion in Teile-undherrsche-Ansätzen akkurate Polynomialzeitmethoden benötigt. Diese Arbeit beschäftigt sich mit der akkuraten Rekonstruktion von Superbäumen in Polynomialzeit. Wir präsentieren Bad Clade Deletion (BCD), eine neue Polynomialzeitheuristik zur Superbaumrekonstruktion. BCD verwendet minimale Schnitte in Graphen, um eine minimale Anzahl von Spalten aus der Matrixrepräsentation zu löschen, sodass diese konfliktfrei wird. Im Gegensatz zu lokalen Suchheuristiken garantiert BCD die Rekonstruktion einer perfekten Phylogenie, sofern eine solche für die Eingabematrix existiert. BCD ermöglicht es, Gütekriterien der Eingabebäume zu berücksichtigen, ohne dass sich dadurch die Komplexität erhöht. Weiterhin zeigen wir, wie zuverlässige Kladen verwendet werden können, um den Suchraum für BCD einzuschränken und wie man diese mit Hilfe des Greedy Strict Consensus Mergers aus den Eingabedaten gewinnen kann. Schließlich stellen wir eine Strahlensuche für BCD vor. Diese erlaubt es eine bestimmte Anzahl suboptimaler Teillösungen (anstatt nur der optimalen) zu berücksichtigen, um so das Gesamtergebnis zu verbessern. Die Worst-Case-Laufzeit der Strahlensuche ist immer noch polynomiell. Zur Berechnung suboptimaler Teillösungen stellen wir einen exakten und einen randomisierten Algorithmus vor. In einer ausführlichen Evaluation auf mehreren simulierten und biologischen Datensätzen vergleichen wir BCD mit einer repräsentativen Auswahl an Superbaummethoden. Wir haben herausgefunden, dass BCD bei Verwendung von Gütekriterien und Suchraumbeschränkung auf simulierten Daten genauer ist als die akkuratesten evaluierten Superbaummethoden. Gleichzeitig ist BCD deutlich schneller als alle evaluierten Methoden. Die Strahlensuche verbessert die Qualität der BCD-Bäume auf allen Datensätzen, allerdings auf Kosten der Laufzeit. Weiterhin fanden wir heraus, dass ein BCD-Superbaum, der als Startbaum verwendet wird, die Qualität einer Maximum-Likelihood-Baumrekonstruktion verbessern kann. Außerdem kann BCD Datensätze verarbeiten, die so groß sind, dass lokale Suchheuristiken auf diesen nicht mehr in angemessener Zeit konvergieren. Aufgrund der Kombination aus Geschwindigkeit, Genauigkeit und der Fähigkeit, den Elternbaum zu rekonstruieren, sofern ein solcher existiert, ist BCD ein vielversprechender Ansatz um die Skalierbarkeit von Teile-und-herrsche-Methoden entscheidend zu verbessern

Algorithms for efficient phylogenetic tree construction

The rapidly increasing amount of available genomic sequence data provides an abundance of potential information for phylogenetic analyses. Many models and methods have been developed to build evolutionary trees based on this information. A common feature of most of these models is that they start out with fragments of the genome, called genes. Depending on the genes and species, and the methods used to perform the phylogenetic analyses, one typically ends up with a large number of phylogenetic trees which may not agree with one another. Simply put, the problem now is the following: Given several discordant phylogenetic trees as input, infer the (presumably) correct phylogeny. This thesis seeks to address some of the methodological and algorithmic challenges posed by this problem. In particular, we present two new algorithms related to inferring phylogenetic trees in the presence of gene duplication, and introduce a new distance measure for comparing phylogenetic trees