J.C.M. Keijsper

Constructing Level-2 Phylogenetic Networks from Triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so, to construct such a network. Here, we extend this work by showing that this problem is even polynomial time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily nontree-like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also implemented the algorithm and applied it to yeast data.

Constructing level-2 phylogenetic networks from triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so to construct such a network [18]. Here we extend this work by showing that this problem is even polynomial-time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily non-tree like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also implemented the algorithm and applied it to yeast data.

An Efficient Algorithm for Minimum-Weight Bibranching

Given a directed graphD=(V,A) and a setS?V, a bibranching is a set of arcsB?Athat contains av?(V\S) path for everyv?Sand anS?vpath for everyv?V\S. In this paper, we describe a primal?dual algorithm that determines a minimum weight bibranching in a weighted digraph. It has running timeO(n?(m+nlogn)), wherem=|A|,n=|V| andn?=min{|S|,|V\S|}. Thus, our algorithm obtains the best known bounds for two important special cases of the problem: bipartite edge cover andr-branching.

Beaches of islands of tractability: algorithms for parsimony and minimum perfect phylogeny haplotyping problems

The problem Parsimony Haplotyping (PH) asks for the smallest set of haplotypes which can explain a given set of genotypes, and the problem Minimum Perfect Phylogeny Haplotyping (MPPH) asks for the smallest such set which also allows the haplotypes to be embedded in a perfect phylogeny evolutionary tree, a well-known biologically-motivated data structure. For PH we extend recent work of [17] by further mapping the interface between “easy” and “hard” instances, within the framework of (k,l)-bounded instances. By exploring, in the same way, the tractability frontier of MPPH we provide the first concrete, positive results for this problem, and the algorithms underpinning these results offer new insights about how MPPH might be further tackled in the future. In both PH and MPPH intriguing open problems remain.

On Packing Connectors

Given an undirected graphG=(V,E) and a partition {S,T} ofV, anS?Tconnector is a set of edgesF?Esuch that every component of the subgraph (V,F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |?G(V1)????G(Vt)|?ktfor every collection {V1,?,Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of Konig on disjoint edge covers in a bipartite graph.