Treewidth-based algorithms for the small parsimony problem on networks

Phylogenetic networks are used to display the relationship among 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 Multiterminal Cut, 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 the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available

Phylogenetic networks are powerful models for representing evolutionary history in studies of genome evolution involving reticulation events. The complex structure of phylogenetic networks makes their inference challenging, especially for large numbers of taxa. To better understand the full space of phylogenetic networks, we derive closed-form formulas for counting binary phylogenetic networks with two and three reticulation events. In addition, we also use our approach to give a simple proof of the known asymptotic result for a fixed number of reticulations.

Reconstructing unobserved ancestral states of a phylogenetic tree provides insight into the history of evolving systems and is one of the fundamental problems in phylogenetics. For a fixed phylogenetic tree, the most parsimonious ancestral reconstruction - a solution to the small parsimony problem - can be efficiently found using the dynamic programming algorithms of Fitch-Hartigan and Sankoff. Ancestral reconstruction is important in many applications including inferring the routes of metastases in cancer, deriving the transmission history of viruses, determining the direction of cellular differentiation in organismal development, and detecting recombination and horizontal gene transfer in phylogenetic networks. However, most of these applications impose additional global constraints on the reconstructed ancestral states, which break the local structure required in the recurrences of Fitch-Hartigan and Sankoff. We introduce an alternative, polyhedral approach to ancestral reconstruction problems using the tree labeling polytope, a geometric object whose vertices represent the feasible ancestral labelings of a tree. This framework yields a polynomial-time linear programming algorithm for the small parsimony problem. More importantly, the tree labeling polytope facilitates the incorporation of additional constraints that arise in modern ancestral reconstruction problems. We demonstrate the utility of our approach by deriving mixed-integer programming algorithms with a small number of integer variables and strong linear relaxations for three such problems: the parsimonious migration history problem, the softwired small parsimony problem on phylogenetic networks, and the convex recoloring problem on trees. Our algorithms outperform existing state-of-the-art methods on both simulated and real datasets. For instance, our algorithm scales to trace routes of cancer metastases in trees with thousands of leaves, enabling the analysis of large trees generated by recent single-cell sequencing technologies. On a mouse model of metastatic lung adenocarcinoma, the tree labeling polytope allows us to infer simpler migration histories compared to previous results. Python implementations of the algorithms provided in this work are available at: github.com/raphael-group/tree-labeling-polytope.

The evolutionary history of a set of species is usually described by a rooted phylogenetic tree. Although it is generally undisputed that bifurcating speciation events and descent with modifications are major forces of evolution, there is a growing belief that reticulate events also have a role to play. Phylogenetic networks provide an alternative to phylogenetic trees and may be more suitable for data sets where evolution involves significant amounts of reticulate events, such as hybridization, horizontal gene transfer, or recombination. In this article, we give an introduction to the topic of phylogenetic networks, very briefly describing the fundamental concepts and summarizing some of the most important combinatorial methods that are available for their computation.

Phylogenetic networks are useful for visualizing evolutionary relationships between species with reticulate events such as hybridizations and horizontal gene transfers. In this paper, we consider the problem of constructing undirected phylogenetic networks that (1) are planar graphs and (2) admit embeddings in the plane where the vertices labeling all taxa are on the boundary of the network. We develop a new algorithm for constructing phylogenetic networks satisfying these constraints. First, we show that only approximate networks can be constructed for some distance matrices with at least five taxa. Then we prove that any five-point metric can be represented approximately by a planar boundary-labeled network with guaranteed fit value of 94.79. We extend the networks constructed in the proof to design an algorithm for computing planar boundary-labeled networks for any number of taxa.

Conjunctive queries ( CQ s) have been widely used in database systems in which acyclic CQ s can be computed efficiently, whereas cyclic CQ s may not. Here, a CQ is acyclic if its hypergraph representation H is acyclic. In order to find a class of CQ s that are "mildly cyclic", hypertree decompositions (HDs) have been studied. The quality of such HDs is by the so-called hypertree width. The class of acyclic queries is the queries whose hypertree width is 1, and a mildly cyclic CQ can be processed efficiently if its hypertree width is bounded. There are several HDs, such as tree decomposition (TD), generalized hypertree decomposition (GHD), fractional hypertree decomposition (FHD), as well as hypertree decomposition (HD). The minimum hypertree width by FHD is the smallest among all, and it is NP-complete to check if the minimum hypertree width by FHD exists for a given hypertree width at most k. In the literature, there is no dynamic programming ( DP ) algorithm or branch-&-bound algorithm reported to compute FHD. In this paper, we show that there is a DP algorithm for FHD, and we give a branch-&-bound algorithm based on our DP algorithm to compute FHD with upper/lower bounds. We confirm the effectiveness and efficiency of our algorithm by testing all 3,648 hypergraphs given in a benchmark for HDs, and we also confirm our approach in query evaluation in real database systems.

With the number of sequenced genomes growing ever larger, it is now common practice to concatenate sequence alignments from several genomic loci as a first step to phylogenetic tree inference. However, as different loci may support different trees due to processes such as gene duplication and lineage sorting, it is important to better understand how commonly used phylogenetic inference methods behave on such "phylogenetic mixtures". Here we shall focus on how parsimony, one of the most popular methods for reconstructing phylogenetic trees, behaves for mixtures of two trees. In particular, we show that (i) the parsimony problem is NP-complete for mixtures of two trees, (ii) there are mixtures of two trees that have exponentially many (in the number of leaves) most parsimonious trees, and (iii) give an explicit description of the most parsimonious tree(s) and scores corresponding to the mixture of a pair of trees related by a single TBR operation.

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof [18] such that the dynamic programming algorithm runs in time O *(2 h ), where h is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph [21]. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.

D-FLAT is a framework for developing algorithms that solve computational problems by dynamic programming on a tree decomposition of the problem instance. The dynamic programming algorithm is specified by means of Answer-Set Programming ASP, allowing for declarative and succinct specifications. D-FLAT traverses the tree decomposition and calls an ASP system with the provided specification at each tree decomposition node. It is thus crucial that the evaluation of the ASP program is done in an efficient way. As experiments have shown, problems that include weights or more involved arithmetics slow down this step significantly due to the grounding step in ASP, which yields large ground programs in these cases. To overcome this problem, we equip D-FLAT with built-in counters in order to shift certain computations from the ASP side to the internal part of D-FLAT. In this paper, we highlight this new feature and provide empirical benchmarks on weighted versions of the Dominating Set problem showing that our new version increases D-FLAT's robustness and efficiency.

The multispecies coalescent (MSC) is a statistical framework that models how gene genealogies grow within the branches of a species tree. The field of computational phylogenetics has witnessed an explosion in the development of methods for species tree inference under MSC, owing mainly to the accumulating evidence of incomplete lineage sorting in phylogenomic analyses. However, the evolutionary history of a set of genomes, or species, could be reticulate due to the occurrence of evolutionary processes such as hybridization or horizontal gene transfer. We report on a novel method for Bayesian inference of genome and species phylogenies under the multispecies network coalescent (MSNC). This framework models gene evolution within the branches of a phylogenetic network, thus incorporating reticulate evolutionary processes, such as hybridization, in addition to incomplete lineage sorting. As phylogenetic networks with different numbers of reticulation events correspond to points of different dimensions in the space of models, we devise a reversible-jump Markov chain Monte Carlo (RJMCMC) technique for sampling the posterior distribution of phylogenetic networks under MSNC. We implemented the methods in the publicly available, open-source software package PhyloNet and studied their performance on simulated and biological data. The work extends the reach of Bayesian inference to phylogenetic networks and enables new evolutionary analyses that account for reticulation.

Axillary nerve block for a wrist fracture in a patient with facioscapulohumeral (Landouzy-Dejerine) disease

Recently, new techniques have been introduced to speed up dynamic programming algorithms on tree decompositions for connectivity problems: the 'Cut and Count' method and a method called the rank-based approach, based on representative sets and Gaussian elimination. These methods respectively give randomised and deterministic algorithms that are single exponential in the treewidth, and polynomial, respectively linear in the number of vertices. In this paper, we adapt these methods to branch decompositions yielding algorithms, both randomised and deterministic, that are in many cases faster than when tree decompositions would be used. In particular, we obtain the currently fastest randomised algorithms for several problems on planar graphs. When the involved weights are O(n^{O(1)}), we obtain faster randomised algorithms on planar graphs for Steiner Tree, Connected Dominating Set, Feedback Vertex Set and TSP, and a faster deterministic algorithm for TSP. When considering planar graphs with arbitrary real weights, we obtain faster deterministic algorithms for all four mentioned problems.

Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. Usually, implementations rely on tables for storing information, and algorithms specify how tuples are manipulated during traversal of the decomposition. However, a bottleneck of such table-based algorithms is relatively high memory consumption. Binary Decision Diagrams (BDDs) and related concepts have been shown to be very well suited to store information efficiently. While several techniques have been proposed that combine DP with efficient BDD-based storage for some particular problems, in this work we present a general approach where DP algorithms are specified on a logical level in form of set-based formula manipulation operations that are executed directly on the BDD data structure. In the paper, we provide several case studies in order to illustrate the method at work, and report on preliminary experiments. These show promising results, both with respect to memory and run-time.

An optimization model for solving the route clustering problem

Algorithmic aspects for multiple-choice hardware/software partitioning