Log-Concave Polynomials IV: Approximate Exchange, Tight Mixing Times, and Near-Optimal Sampling of Forests

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 $O(k\log k)$. Our bound has no dependence on $n$ or the starting point, unlike the previous analyses [ALOV19,CGM19], 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 function $\mu: {[n] \choose k} \to \mathbb{R}{\geq 0},$ our approximate exchange property implies that a simple local search algorithm gives a $k^{{O(k)}$-approximation} of $\max{S} \mu(S)$ when $\mu$ is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when $\mu$ 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 $O(n\log² n).$ The best known result for sampling random forest was a FPAUS with high polynomial runtime recently found by \cite{ALOV19, CGM19}. For spanning tree, we improve on the almost-linear time algorithm by [Sch18]. Our analysis works on weighted graphs too, and is the first to achieve nearly-linear running time for these problems.