Georgios Stamoulis

On unrooted and root-uncertain variants of several well-known phylogenetic network problems

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the “root uncertain” variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

Approximation Algorithms for Bounded Color Matchings via Convex Decompositions

We study the following generalization of the maximum matching problem in general graphs: Given a simple non-directed graph G = (V,E) and a partition of the edges into k classes (i.e. E = E 1 ∪ ⋯ ∪ E k ), we would like to compute a matching M on G of maximum cardinality or profit, such that |M ∩ E j | ≤ w j for every class E j . Such problems were first studied in the context of network design in [17]. We study the problem from a linear programming point of view: We provide a polynomial time (frac{1}{2})-approximation algorithm for the weighted case, matching the integrality gap of the natural LP formulation of the problem. For this, we use and adapt the technique of approximate convex decompositions [19] together with a different analysis and a polyhedral characterization of the natural linear program to derive our result. This improves over the existing (frac{1}{2}), but with additive violation of the color bounds, approximation algorithm [14].

A note on convex characters, Fibonacci numbers and exponential-time algorithms

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species (taxa) and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. Given an unrooted, binary phylogenetic tree T on a set of n >= 2 taxa, a closed (but fairly opaque) expression for the number of convex characters on T has been known since 1992, and this is independent of the exact topology of T. In this note we prove that this number is actually equal to the (2n-1)th Fibonacci number. Next, we define g_k(T) to be the number of convex characters on T in which each state appears on at least k taxa. We show that, somewhat curiously, g_2(T) is also independent of the topology of T, and is equal to to the (n-1)th Fibonacci number. As we demonstrate, this topological neutrality subsequently breaks down for k >= 3. However, we show that for each fixed k >= 1, g_k(T) can be computed in O(n) time and the set of characters thus counted can be efficiently listed and sampled. We use these insights to give a simple but effective exact algorithm for the NP-hard maximum parsimony distance problem that runs in time

A 0.821-ratio purely combinatorial algorithm for maximum k-vertex cover in bipartite graphs

Theta( phi^{n} cdot n^2 )

On a Fixed Haplotype Variant of the Minimum Error Correction Problem

, where

Treewidth distance on phylogenetic trees

phi approx 1.618...

The multi-budgeted and weighted bounded degree metric Steiner network problem

is the golden ratio, and an exact algorithm which computes the tree bisection and reconnection distance (equivalently, a maximum agreement forest) in time

Purely combinatorial approximation algorithms for maximum k-vertex cover in bipartite graphs

Theta( phi^{2n}cdot text{poly}(n))

The Computational Complexity of Stochastic Optimization

, where

On the approximation of maximum

phi^2 approx 2.619