In this work we refine the analysis of the distributed Laplacian solver recently established by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS 21), via the Ghaffari-Haeupler framework (SODA 16) of low-congestion shortcuts. Specifically, if $epsilon > 0$ represents the error of the solver, we derive two main results. First, for any $n$-node graph $G$ with hop-diameter $D$ and treewidth bounded by $k$, we show the existence of a Laplacian solver with round complexity $O(n^{o(1)}kD log(1/epsilon))$ in the CONGEST model. For graphs with bounded treewidth this circumvents the notorious $Omega(sqrt{n})$ lower bound for global problems in general graphs. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with very limited global power in the form of the recently introduced node-capacitated clique. In this model, we show the existence of a Laplacian solver with round complexity $O(n^{o(1)} log(1/epsilon))$. The unifying thread of these results is an application of accelerated distributed algorithms for a congested variant of the standard part-wise aggregation problem that we introduce. This primitive constitutes the primary building block for simulating local operations on low-congestion minors, and we believe that this framework could be more generally applicable.
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