Log-concave polynomials IV: approximate exchange, tight mixing times, and near-optimal sampling of forests

Abstract

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.