Georgia Institute of Technology College of Sciences

Motivated by real-time logistics, I will present a deterministic online algorithm for the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given a set S of server locations and a set R of request locations.The requests arrive one at a time and when it arrives, we must immediately and irrevocably match it to a ``free" server. The cost of matching a server to request is given by the distance between the two locations (which we assume satisfies triangle inequality). The objective of this problem is to come up with a matching of servers to requests which is competitive with respect to the minimum-cost matching of S and R.In this talk, I will show that this new deterministic algorithm performs optimally across different adversaries and metric spaces. In particular, I will show that this algorithm simultaneously achieves optimal performances in two well-known online models -- the adversarial and the random arrival models. Furthermore, the same algorithm also has an exponentially improved performance for the line metric resolving a long standing open question.

A graph is ``strongly regular'' (SRG) if it is $k$-regular, and every pair of adjacent (resp. nonadjacent) vertices has exactly $\lambda$ (resp. $\mu$) common neighbors. Paradoxically, the high degree of regularity in SRGs inhibits their symmetry. Although the line-graphs of the complete graph and complete bipartite graph give examples of SRGs with $\exp(\Omega(\sqrt{n}))$ automorphisms, where $n$ is the number of vertices, all other SRGs have much fewer---the best bound is currently $\exp(\tilde{O}(n^{9/37}))$ (Chen--Sun--Teng, 2013), and Babai conjectures that in fact all primitive SRGs besides the two exceptional line-graph families have only quasipolynomially-many automorphisms. In joint work with Babai, Chen, Sun, and Teng, we make progress toward this conjecture by giving a quasipolynomial bound on the number of automorphisms for valencies $k > n^{5/6}$. Our proof relies on bounds on the vertex expansion of SRGs to show that a polylogarithmic number of randomly chosen vertices form a base for the automorphism group with high probability.

We consider the extremal counting problem which asks what d-regular, r-uniform hypergraph on n vertices has the largest number of (strong) independent sets. Our goal is to generalize known results for number of matchings and independent sets in regular graphs to give a general bound in the hypergraph case. In particular, we propose an adaptation to the hypergraph setting of the occupancy fraction method pioneered by Davies et al. (2016) for use in the case of graph matchings. Analysis of the resulting LP leads to a new bound for the case r=3 and suggests a method for tackling the general case.

A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett-Meka algorithm finds a half integral point in any "large enough" polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set K with measure at least exp(-n/500), the following algorithm finds a point y in K \cap [-1,1]^n with Omega(n) coordinates in {-1,+1}: (1) take a random Gaussian vector x; (2) compute the point y in K \cap [-1,1]^n that is closest to x. (3) return y. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a Theorem of Gluskin and Giannopoulos.