Georgia Institute of Technology College of Sciences

We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a $(1 \pm \epsilon)$-spectral sparsifier with amortized update time $poly(\log{n},\epsilon^{-1})$. Second, we give a fully dynamic algorithm for maintaining a $(1 \pm \epsilon)$-cut sparsifier with worst-case update time $poly(\log{n},\epsilon^{-1})$. Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a $(1 + \epsilon)$-approximate minimum cut in an unweighted, undirected, bipartite graph with amortized update time $poly(\log{n},\epsilon^{-1})$.Joint work with Ittai Abraham, Ioannis Koutis, Sebastian Krinninger, and Richard Peng

A special feature possessed by the graphs of social networks is triangle-dense. R. Gupta, T. Roughgarden and C. Seshadhri give a polynomial time graph algorithm to decompose a triangle-dense graph into some clusters preserving high edge density and high triangle density in each cluster with respect to the original graph and each cluster has radius 2. And high proportion of triangles of the original graph are preserved in these clusters. Furthermore, if high proportion of edges in the original graph is "locally triangle-dense", then additionally, high proportion of edges of the original graph are preserved in these clusters. In this talk, I will present most part of the paper "Decomposition of Triangle-dense Graphs" in SIAM J. COMPUT. Vol. 45, No. 2, pp. 197–215, 2016, by R. Gupta, T. Roughgarden and C. Seshadhri.

Abstract: Certain materials form geometric structures called "grains," which means that one has distinct volumes filled with the same semi-solid material but not mixing. This can happen with semi-molten copper and something like this can also happen with liquid crystals (which are used in some calculator display screens). People who try to analyze such systems tend to be interested in the motion of the boundaries between grains (which are often modeled by mean curvature flow) and the motions of the exterior surfaces of grains (which are often modeled by surface diffusion flow). Surfaces of constant mean curvature are stationary for both flows and provide stationary or equilibrium configurations. The surfaces of constant mean curvature which are axially symmetric have been classified. Grain boundaries are not usually axially symmetric, but I will describe a model situation in which they are and one can study the resulting equilibria. I will give a very informal introduction to the flow problems mentioned above (about which I know very little) and then go over the classification of axially symmetric constant mean curvature surfaces (about which I know rather more) and some reasonable questions one can ask (and hopefully answer) about such problems.

The Frankl union-closed sets conjecture states that there exists an element present in at least half of the sets forming a union-closed family. We reformulate the conjecture as an optimization problem and present an integer program to model it. The computations done with this program lead to a new conjecture: we claim that the maximum number of sets in a non-empty union-closed family in which each element is present at most a times is independent of the number n of elements spanned by the sets if n is greater or equal to log_2(a)+1. We prove that this is true when n is greater or equal to a. We also discuss the impact that this new conjecture would have on the Frankl conjecture if it turns out to be true. This is joint work with Jonad Pulaj and Dirk Theis.