We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank k on a ground set of n elements, or more generally distributions associated with log-concave polynomials of homogeneous degree k on n variables, we show that the down-up random walk, started from an arbitrary point in the support, mixes in time O(klogk). Our bound has no dependence on n or the starting point, unlike the previous analyses of Anari et al. (STOC 2019), Cryan et al. (FOCS 2019), and is tight up to constant factors. The main new ingredient is a property we call approximate exchange, a generalization of well-studied exchange properties for matroids and valuated matroids, which may be of independent interest. In particular, given a distribution µ over size-k subsets of [n], our approximate exchange property implies that a simple local search algorithm gives a kO(k)-approximation of maxS µ(S) when µ is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when µ is strongly Rayleigh. As an application, we show how to leverage down-up random walks to approximately sample random forests or random spanning trees in a graph with n edges in time O(nlog2 n). The best known result for sampling random forest was a FPAUS with high polynomial runtime recently found by Anari et al. (STOC 2019), Cryan et al. (FOCS 2019). For spanning tree, we improve on the almost-linear time algorithm by Schild (STOC 2018). Our analysis works on weighted graphs too, and is the first to achieve nearly-linear running time for these problems. Our algorithms can be naturally extended to support approximately sampling from random forests of size between k1 and k2 in time O(n log2 n), for fixed parameters k1, k2.

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0<q<1. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least 1.

We give a deterministic polynomial time 2^O(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 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, and Katz and Huh and Wang on combinatorial hodge theory to derive a connection between matroids and log-concave polynomials. We expect that several new applications in approximation algorithms will be derived from this connection in future. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is log-concave as a function 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 corresponding polynomials.