Minor Sparsifiers and the Distributed Laplacian Paradigm

Forster, S., Goranci, G., Liu, Y. P., Peng, R., Sun, X. and Ye, M. (2022) Minor Sparsifiers and the Distributed Laplacian Paradigm. In: 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), Denver, CO, USA, 07-10 February 2022, pp. 989-999. ISBN 9781665420563 (doi: 10.1109/focs52979.2021.00099)

Abstract

We study distributed algorithms built around minor-based vertex sparsifiers, and give the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy. Our Laplacian solver has a round complexity of $O(n^{o(1)}(\sqrt{n}+D))$, and thus almost matches the lower bound of $\widetilde{\Omega}(\sqrt{n}+D)$, where $n$ is the number of nodes in the network and $D$ is its diameter. We show that our distributed solver yields new sublinear round algorithms for several cornerstone problems in combinatorial optimization. This is achieved by leveraging the powerful algorithmic framework of Interior Point Methods (IPMs) and the Laplacian paradigm in the context of distributed graph algorithms, which entails numerically solving optimization problems on graphs via a series of Laplacian systems. Problems that benefit from our distributed algorithmic paradigm include exact mincost flow, negative weight shortest paths, maxflow, and bipartite matching on sparse directed graphs. For the maxflow problem, this is the first exact distributed algorithm that applies to directed graphs, while the previous work by [Ghaffari et al. SICOMP'18] considered the approximate setting and works only for undirected graphs. For the mincost flow and the negative weight shortest path problems, our results constitute the first exact distributed algorithms running in a sublinear number of rounds. Given that the hybrid between IPMs and the Laplacian paradigm has proven useful for tackling numerous optimization problems in the centralized setting, we believe that our distributed solver will find future applications. At the heart of our distributed Laplacian solver is the notion of spectral subspace sparsifiers of [Li, Schild FOCS'18]. We present a nontrivial distributed implementation of their construction by (i) giving a parallel variant of their algorithm that avoids the sampling of random spanning trees and uses approximate leverage scores instead, and (ii) showing that the algorithm still produces a high-quality subspace spectral sparsifier by carefully setting up and analyzing matrix martingales. Combining this vertex reduction recursively with both tree and elimination-based preconditioners leads to our algorithm for solving Laplacian systems. The construction of the elimination-based preconditioners is based on computing short random walks, and we introduce a new technique for reducing the congestion incurred by the simulation of these walks on weighted graphs.

• Item Type: Conference Proceedings
• Additional Information: Sebastian Forster is supported by the Austrian Science Fund (FWF): P 32863-N. Part of this work was done while Gramoz Goranci was a postdoc at University of Toronto, and supported by Sushant Sachdeva’s Discovery grant from NSERC. Yang P. Liu was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. Richard Peng is supported by the National Science Foundation (NSF) under Grant No. 1846218. Xiaorui Sun is supported by start-up funds from University of Illinois at Chicago.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Goranci, Dr Gramoz
• Authors: Forster, S., Goranci, G., Liu, Y. P., Peng, R., Sun, X., and Ye, M.
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
• Publisher: IEEE
• ISSN: 1523-8288
• ISBN: 9781665420563
• Copyright Holders: Copyright © 2022 IEEE
• First Published: First published in Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS): 989-999
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher

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