1 November 2021 Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids
Nima Anari, Shayan Oveis Gharan, Cynthia Vinzant
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Duke Math. J. 170(16): 3459-3504 (1 November 2021). DOI: 10.1215/00127094-2020-0091


We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid.

There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.


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Nima Anari. Shayan Oveis Gharan. Cynthia Vinzant. "Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids." Duke Math. J. 170 (16) 3459 - 3504, 1 November 2021. https://doi.org/10.1215/00127094-2020-0091


Received: 6 August 2019; Revised: 19 June 2020; Published: 1 November 2021
First available in Project Euclid: 11 October 2021

MathSciNet: MR4332671
zbMATH: 1507.68345
Digital Object Identifier: 10.1215/00127094-2020-0091

Primary: 68W25
Secondary: 90C25

Keywords: Approximate counting , combinatorial Hodge theory , Convex programming , log-concave polynomials , matroid intersection , matroids

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 16 • 1 November 2021
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