This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network $N$ can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of $N$ and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

Abstract Dynamic Time Warping (DTW) is a well-known similarity measure for time series. The standard dynamic programming approach to compute the DTW distance of two length-n time series, however, requires $$O(n^2)$$ O ( n 2 ) time, which is often too slow for real-world applications. Therefore, many heuristics have been proposed to speed up the DTW computation. These are often based on lower bounding techniques, approximating the DTW distance, or considering special input data such as binary or piecewise constant time series. In this paper, we present a first exact algorithm to compute the DTW distance of two run-length encoded time series whose running time only depends on the encoding lengths of the inputs. The worst-case running time is cubic in the encoding length. In experiments we show that our algorithm is indeed fast for time series with short encoding lengths.

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.

Phylogenetic reconstruction is one of the paramount challenges of contemporary bioinformatics. A subtask of existing tree reconstruction algorithms is modeled by the Small Parsimony problem: given a tree T and an assignment of character-states to its leaves, assign states to the internal nodes of T such as to minimize the parsimony score, that is, the number of edges of T connecting nodes with different states. While this problem is polynomial-time solvable on trees, the matter is more complicated if T contains reticulate events such as hybridizations or recombinations, i.e. when T is a network. Indeed, three different versions of the parsimony score on networks have been proposed and each of them is NP-hard to decide. Existing parameterized algorithms focus on combining the number of possible character-states with the number of reticulate events (per biconnected component). Here, we consider the treewidth of the undirected graph underlying the input network as parameter, presenting dynamic programming algorithms for (slight generalizations of) all three versions of the parsimony problem on networks. Our algorithms use a formulation of the treewidth that may facilitate formalizing treewidth-based dynamic programming algorithms on phylogenetic networks for other problems.

Scaffolding is the final step in assembling Next Generation Sequencing data, in which pre-assembled contiguous regions (”contigs”) are oriented and ordered using information that links them (for example, mapping of paired-end reads). As the genome of some species is highly repetitive, we allow placing some contigs multiple times, thereby generalizing established computational models for this problem. We study the subsequent problems induced by the translation of solutions of the model back to actual sequences, proposing models and analyzing the complexity of the resulting computational problems. We find both polynomial-time and $$\mathcal {NP}$$ -hard special cases like planarity or bounded degree. Finally, we propose two polynomial-time approximation algorithms according to cut/weight score.

We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately. In particular, we show that it is NP-hard to decide (a) whether there is some common index at which all given periodic, binary sequences are 0, and (b) whether a single cop can catch a single robber on an edge-periodic temporal graph. We further present results for various parameterizations of both problems and show that hardness not only applies in general, but also for highly limited instances. As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W[1]-hard when parameterized by the size of the underlying input graph.

In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition.

Summary: Gene trees reconstructed from sequence alignments contain poorly supported branches when the phylogenetic signal in the sequences is insufficient to determine them all. When a species tree is available, the signal of gains and losses of genes can be used to correctly resolve the unsupported parts of the gene history. However finding a most parsimonious binary resolution of a non-binary tree obtained by contracting the unsupported branches is NP-hard if transfer events are considered as possible gene scale events, in addition to gene origination, duplication and loss. We propose an exact, parameterized algorithm to solve this problem in single-exponential time, where the parameter is the number of connected branches of the gene tree that show low support from the sequence alignment or, equivalently, the maximum number of children of any node of the gene tree once the low-support branches have been collapsed. This improves on the best known algorithm by an exponential factor. We propose a way to choose among optimal solutions based on the available information. We show the usability of this principle on several simulated and biological datasets. The results are comparable in quality to several other tested methods having similar goals, but our approach provides a lower running time and a guarantee that the produced solution is optimal. Availability and Implementation: Our algorithm has been integrated into the ecceTERA phylogeny package, available at http://mbb.univ-montp2.fr/MBB/download_sources/16__ecceTERA and which can be run online at http://mbb.univ-montp2.fr/MBB/subsection/softExec.php?soft=eccetera . Contact: [email protected]. Supplementary information: Supplementary data are available at Bioinformatics online.

This paper presents new structural and algorithmic results around the scaffolding problem, which occurs prominently in next generation sequencing. The problem can be formalized as an optimization problem on a special graph, the "scaffold graph". We prove that the problem is polynomial if this graph is a tree by providing a dynamic programming algorithm for this case. This algorithm serves as a basis to deduce an exact algorithm for general graphs using a tree decomposition of the input. We explore other structural parameters, proving a linear-size problem kernel with respect to the size of a feedback-edge set on a restricted version of Scaffolding. Finally, we examine some parameters of scaffold graphs, which are based on real-world genomes, revealing that the feedback edge set is significantly smaller than the input size.

The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set $V$ and a collection of subsets $V_1, V_2, \ldots, V_m \subseteq V$, and asks for a minimum-cardinality edge set $E$ such that for the graph $G=(V,E)$ all induced subgraphs $G[V_1], G[V_2], \ldots, G[V_m]$ are connected. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number $m$ of subsets, implying fixed-parameter tractability for the parameter $m$. Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate on polynomial-time data reduction rules which also may be of practical use in solving Subset Interconnection Design.

The aim of directed Eulerian extension problems is to make a given directed, possibly arc-weighted, (multi-)graph Eulerian by adding a minimum-cost set of arcs. These problems have natural applications in scheduling and arc routing and are closely related to the Chinese Postman and Rural Postman problems. Our main result is to show that the NP-hard Weighted Multigraph Eulerian Extension problem is fixed-parameter tractable with respect to the number ${k}$ of extension arcs. For a directed $n$-vertex multigraph, the corresponding running time amounts to ${\ensuremath{O(4^{k}\cdot n^3)}}$. This also implies a fixed-parameter tractability result for the “equivalent” Rural Postman problem parameterized above guarantee. In addition, we present several polynomial-time algorithms for natural Eulerian extension problems, including undirected variants which can be defined analogously to the directed ones.