Georgia Institute of Technology College of Sciences

As a generalization of many classic problems in combinatorial optimization, submodular optimization has found a wide range of applications in machine learning (e.g., in feature engineering and active learning). For many large-scale optimization problems, we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient for a (distributed) algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by such applications, we study the adaptivity and query complexity of adaptive submodular optimization. Our main result is a distributed algorithm for maximizing a monotone submodular function with cardinality constraint $k$ that achieves a $(1-1/e-\varepsilon)$-approximation in expectation. Furthermore, this algorithm runs in $O(\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation oracle in expectation. All three of these guarantees are optimal, and the query complexity is substantially less than in previous works. Finally, to show the generality of our simple algorithm and techniques, we extend our results to the submodular cover problem. Joint work with Vahab Mirrokni and Morteza Zadimoghaddam (arXiv:1807.07889).

We will give a brief introduction to matroids with a focus on representable matroids. We will also discuss the Plücker embedding of the Grassmannian.

An $r$-cut of a $k$-uniform hypergraph $H$ is a partition of the vertex set of $H$ into $r$ parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every $m$-edge graph has a 2-cut of size $m/2+\Omega(\sqrt{m})$, and this is best possible. In this talk we will discuss recent results on analogues of Edwards’ result and related problems in hypergraphs.

In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.