Computer Science/Discrete Mathematics Seminar I

Topic: Strongly log concave polynomials, high dimensional simplicial complexes, and an FPRAS for counting Bases of Matroids

Speaker: Shayan Oveis Gharan

Affiliation: University of Washington

Date & Time: Monday February 25th, 2019, 11:00am - 12:00pm

Location: Simonyi Hall 101

Video: https://video.ias.edu/csdm/2019/0225-ShayanOveisGharan

A matroid is an abstract combinatorial object which generalizes the notions of spanning trees, and linearly independent sets of vectors. I will talk about an efficient algorithm based on the Markov Chain Monte Carlo technique to approximately count the number of bases of any given matroid. The proof is based on a new connections between high dimensional simplicial complexes, and a new class of multivariate polynomials called completely log-concave polynomials. In particular, we exploit a fundamental fact from our previous work that the bases generating polynomial of any given matroid is a log-concave function over the positive orthant. Based on joint works with Nima Anari, Kuikui Liu, and Cynthia Vinzant.