COMS

This workshop will bring together researchers from these different fields. The goal is to explore connections, learn and share techniques, and build bridges.

This workshop will be an opportunity for researchers from diverse fields to get together and share problems and techniques for handling and analyzing graphs structures. The connections---mathematical, computational, and practical---that arise between these seemingly-diverse problems and approaches will be emphasized.

However, while the eigenvalues and eigenvectors of the Laplacian capture a striking amount of the structure of the graph, they certainly do not capture all of it. Recent work in the field suggests that we have only scratched the surface of what can be done if we are willing to broaden our investigation to include more general linear-algebraic properties of the matrices we associate to graphs.

A particularly fruitful example of this has been the study of Laplacian linear systems, where the interplay between linear algebra and graph theory has led to progress in both fields. On the one hand, researchers have used the combinatorial structure of the corresponding graphs to facilitate the solution of these linear systems, resulting in solvers that run in nearly-linear time. On the other hand, one can use these linear systems to describe the behavior of electrical flows on a graph, which has provided a powerful new primitive for algorithmic graph theory. This interaction has already led to improved algorithmic results for many of the basic problems in algorithmic graph theory, including finding maximum flows and minimum cuts, solving traveling salesman problems, sampling random trees, sparsifying graphs, computing multicommodity flows, and approximately solving a wide range of general clustering and partitioning problems. In addition, researchers have recently shown how to exploit a wide range of other algebraic properties of matrices associated to graphs, such as the threshold rank, cut norm, sensitivity to perturbation, or hypercontractivity of the eigenspaces, to achieve impressive algorithmic results.

In this workshop, we will bring researchers together to study and advance this new emerging frontier in algorithmic graph theory.