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Minor Sparsifiers and the Distributed Laplacian Paradigm | IEEE Conference Publication | IEEE Xplore

Minor Sparsifiers and the Distributed Laplacian Paradigm


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 Laplaci...Show More

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) sho...
Date of Conference: 07-10 February 2022
Date Added to IEEE Xplore: 04 March 2022
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Conference Location: Denver, CO, USA

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I. Introduction

The steady growth of data makes it increasingly important to control and reduce the communication of algorithms. The CONGEST model [1] is a widely studied model for low communication algorithms on large graphs and sparse matrices. In this model, each vertex/variable occupies a separate machine, and communicates in synchronous rounds by sending messages of length to its neighbors given by the edges of the underlying graph. This bandwidth restriction implies a polynomial lower bound in the round complexity for many fundamental graph problems [2]–[4]. While early work on efficient algorithms in this model has focused on the minimum spanning tree problem [5]–[7], extensive work over the past few years has led to efficient algorithms for several more fundamental graph problems, such as approximate and exact single-source shortest paths [8]–[13], approximate and exact all-pairs shortest paths [14]–[21], approximate and exact minimum cut [22]–[26], approximate maximum flow [27], bipartite maximum matching [28], triangle counting [29]–[31], and single-source reachability [32], [33].

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