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We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$\footnote{Throughout, $\widetilde{O}(\cdot)$ hides polylogarithmic factors in $n$.} time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. This is the first nearly-linear runtime in the output size, which is clearly optimal. For a determinantal point process on $k$-sized subsets of a ground set of $n$ elements, defined via an $n\times n$ kernel matrix, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. The time to compute just the weight of the output set is simply $\simeq k^\omega$, a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, the domain size can be reduced to nearly linear in the output size $t=\widetilde{O}(k)$, improving the state of the art from $t= \widetilde{O}(k^2)$ for general strongly Rayleigh distributions and the more specialized $t=\widetilde{O}(k^{1.5})$ for spanning tree distributions. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of $\mu$ are known and stored in a convenient data structure. Having access to marginals is the discrete analog of having access to the mean and covariance of a continuous distribution, or equivalently knowing ``isotropy'' for the distribution, the key behind optimal samplers in the continuous setting based on the famous Kannan-Lov\'asz-Simonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures. 

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in m^{1+o(1)} time. Our algorithm builds the flow through a sequence of m^{1+o(1)} approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized m^{o(1)} time using a new dynamic graph data structure. Our framework extends to algorithms running in m^{1+o(1)} time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and p-norm isotonic regression on arbitrary directed acyclic graphs.