Georgia Institute of Technology College of Sciences

This is a summary of result of LUC HILLAIRET AND CHRIS JUDGE.

The goal of this talk is to show recent advances regarding two important mathematical problems. The first one can be straightforwardly formulated in a graph theory language, but can be possibly applied in other fields. The second one was motivated by machine learning applications, but leads to graph theory techniques. The celebrated open conjecture of Erdos and Hajnal from 1989 states that families of graphs not having some given graph H as an induced subgraph contain polynomial-size cliques/stable sets (in the undirected setting) or transitive subsets (in the directed setting). Recent techniques developed over last few years provided the proof of the conjecture for new infinite classes of graphs (in particular the first infinite class of prime graphs). Furthermore, they gave tight asymptotics for the Erdos-Hajnal coefficients for many classes of prime tournaments as well as the proof of the conjecture for all but one tournament on at most six vertices and the proof of the weaker version of the conjecture for trees on at most six vertices. In this part of the talk I will summarize these recent achievements. Structured non-linear graph-based hashing is motivated by applications in neural networks, where matrices of linear projections are constrained to have a specific structured form. This drastically reduces the size of the model and speeds up computations. I will show how the properties of the underlying graph encoding correlations between entries of these matrices (such as its chromatic number) imply the quality of the entire non-linear hashing mechanism. Furthermore, I will explain how general structured matrices that very recently attracted researchers’ attention naturally lead to the underlying graph theory description.

We will discuss an extension of the entropy power inequality in terms of the Renyi entropy to sums of independent random vectors (with densities). Joint work with G. Chistyakov.

Represent a genome with an edge-labelled, directed graph having maximum total degree two. We explore a number of questions regarding genome rearrangement, a common mode of molecular evolution. In the single cut-or-join model for genome rearrangement, a genome can mutate in one of two ways at any given time: a cut divides a degree two vertex into two degree one vertices while a join merges two degree one vertices into one degree two vertex. Fix a set of genomes, each having the same set of edge labels. The number of ways for one genome to mutate into another can be computed in polynomial time. The number of medians can also be computed in polynomial time. While single cut-or-join is, computationally, the simplest mathematical model for genome rearrangement, determining the number of most parsimonious median scenarios remains #P-complete. We will discuss these and other complexity results that arose from an abstraction of this problem. [This is joint work with Istvan Miklos.]