Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts

In this paper, we refine the distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS `21) via the Ghaffari-Haeupler framework (SODA `16) of \emph{low-congestion shortcuts}. Specifically, if $\epsilon > 0$ is the error of the Laplacian solver, we obtain two main results. First, in the supported version of the CONGEST model, we establish an almost \emph{universally optimal} Laplacian solver. Namely, we show that any Laplacian system on an $n$-node graph with \emph{shortcut quality} $\text{SQ}(G)$ can be solved after $n^{o(1)} \text{SQ}(G) \log(1/\epsilon)$ rounds, almost matching our lower bound of $\widetilde{\Omega}(\textrm{SQ}(G))$ rounds on \emph{any graph} $G$. Our techniques also imply almost universally optimal Laplacian solvers in the full generality of CONGEST, conditional on the efficient construction of shortcuts. In particular, they unconditionally imply a novel $D n^{o(1)} \log (1/\epsilon)$ Laplacian solver for excluded-minor graphs with hop-diameter $D$. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique (NCC) model. In this model, we show the existence of a Laplacian solver with round complexity $n^{o(1)} \log(1/\epsilon)$. The unifying thread of these results, and our main technical contribution, is the development of nearly-optimal distributed algorithms for a novel \emph{congested} generalization of the standard \emph{part-wise aggregation} problem. This primitive accelerates the Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye, and we believe it will find further applications in the future.