Mathias Weller

Exact approaches for scaffolding

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.

On the Complexity of Scaffolding Problems: From Cliques to Sparse Graphs

This paper is devoted to new results about the scaffolding problem, an integral problem of genome inference in bioinformatics. The problem consists of finding a collection of disjoint cycles and paths covering a particular graph called the scaffold graph. We examine the difficulty and the approximability of the scaffolding problem in special classes of graphs, either close to trees, or very dense. We propose negative and positive results, exploring the frontier between difficulty and tractability of computing and/or approximating a solution to the problem.

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.

On the fixed parameter tractability of agreement-based phylogenetic distances

Three important and related measures for summarizing the dissimilarity in phylogenetic trees are the minimum number of hybridization events required to fit two phylogenetic trees onto a single phylogenetic network (the hybridization number), the (rooted) subtree prune and regraft distance (the rSPR distance) and the tree bisection and reconnection distance (the TBR distance) between two phylogenetic trees. The respective problems of computing these measures are known to be NP-hard, but also fixed-parameter tractable in their respective natural parameters. This means that, while they are hard to compute in general, for cases in which a parameter (here the hybridization number and rSPR/TBR distance, respectively) is small, the problem can be solved efficiently even for large input trees. Here, we present new analyses showing that the use of the “cluster reduction” rule—already defined for the hybridization number and the rSPR distance and introduced here for the TBR distance—can transform any

Parameterized certificate dispersal and its variants

O(f(p) cdot n)

On the Linearization of Scaffolds Sharing Repeated Contigs

O(f(p)·n)-time algorithm for any of these problems into an

On Residual Approximation in Solution Extension Problems

O(f(k) cdot n)