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

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:Monday, February 25
Time/Room:11:00am - 12:00pm/Simonyi Hall 101
Video Link: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.