Georgia Institute of Technology College of Sciences

Estimating the Tree of Life, an evolutionary tree describing how all life evolved from a common ancestor, is one of the major scientific objectives facing modern biologists. This estimation problem is extremely computationally intensive, given that the most accurate methods (e.g., maximum likelihood heuristics) are based upon attempts to solve NP-hard optimization problems. Most computational biologists assume that the only feasible strategy will involve a divide-and-conquer approach where the large taxon set is divided into subsets, trees are estimated on these subsets, and a supertree method is applied to assemble a tree on the entire set of taxa from the smaller "source" trees. I will present supertree methods in a mathematical context, focusing on some theoretical properties of MRP (Matrix Representation with Parsimony), the most popular supertree method, and SuperFine, a new supertree method that outperforms MRP.

This mini-conference will feature about six speakers on various topics in additive combinatorics.

I will describe some results concerning factorizations ofdiffeomorphisms of compact surfaces with boundary. In particular, Iwill describe a refinement of the well-known \emph{right-veering}property, and discuss some applications to the problem ofcharacterization of geometric properties of contact structures interms of monodromies of supporting open book decompositions.

We consider a the minimum k-way cut problem for unweighted graphs with a bound $s$ on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into k components, or report that this requires cutting more than s edges. We show that this problem is fixed-parameter tractable (FPT) in s. More precisely, for s=O(1), our algorithm runs in quadratic time while we have a different linear time algorithm for planar graphs and bounded genus graphs. Our result solves some open problems and contrasts W[1] hardness (no FPT unless P=NP) of related formulations of the k-way cut problem. Without the size bound, Downey et al.~[2003] proved that the minimum k-way cut problem is W[1] hard in k even for simple unweighted graphs. A simple reduction shows that vertex cuts are at least as hard as edge cuts, so the minimum k-way vertex cut is also W[1] hard in terms of k. Marx [2004] proved that finding a minimum k-way vertex cut of size s is also W[1] hard in s. Marx asked about FPT status with edge cuts, which is what we resolve here. We also survey approximation results for the minimum k-way cut problem, and conclude some open problems. Joint work with Mikkel Thorup (AT&T Research).