Welcome to Combinatorics and Optimization

Title: New Strongly Regular Graphs from Finite Semifields and Finite Geometry

Abstract: Finite geometry often provides natural examples of highly structured combinatorial objects, many of which exhibit strong symmetry properties. 

In particular, many constructions of strongly regular graphs arise from classical geometric configurations. In this talk, we will present two new constructions of quasi-polar spaces, that give rise to two families of pairwise non-isomorphic strongly regular graphs, having the same non-trivial automorphism group.

Both constructions are related to a pair of commuting polarities in a projective space. Surprisingly, one of these constructions is connected to the algebraic structure of finite semifields and their tensor representation.

Title: Log-concavity of random Radon partitions

 Abstract: Over one hundred years ago, Radon proved that any set of d+2 points in R^d can be partitioned into two sets whose convex hulls intersect. I will talk about Radon partitions when the points are selected randomly. In particular, if the points are independent normal random vectors, let p_k be the probability that the Radon partition has size (k, d+2-k). Answering a conjecture of Kalai and White, we show that the sequence (p_k) is ultra log-concave and that, in fact, a balanced partition is the most likely. Joint work with Swee Hong Chan, Gil Kalai, Bhargav Narayanan, and Moshe White.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Exact algorithms for combinatorial interdiction problems

Abstract: Typical optimization paradigms involve a single decision maker that can control all variables involved. However, several practical situations involve multiple (potentially adversarial) decision makers. Bilevel optimization is a field that involves two decision makers. The key paradigm is that an upper-level decision maker acts first and after observing the upper-level decisions, a lower level decision maker optimizes their own objective. One particular important instance of such problems are so-called interdiction problems, where the upper-level decision is to try and forbid access to some decisions by the lower level and the goal of the upper level is to make the most impact into the lower-level decisions. These problems are, in general $\Sigma^P_2$-hard (likely harder than NP-hard). 

In this talk I will present recent work on improving significantly on state-of-the-art exact algorithms to obtain optimal solutions to some combinatorial interdiction problems. Despite the hardness results, our algorithm can solve instances consistently in a short amount of time. We also generalize our algorithms to propose a framework that could be applied to other similar problems, by deriving relaxations based on looking at these problems as games and performing operations on such games. 

This is joint work with Noah Weninger.