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Title:
Strongly log concave polynomials, high dimensional simplicial complexes, and an FPRAS for counting Bases of Matroids
Speaker:
Abstract:
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.
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