Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: patt-sol