The Laplacian Paradigm in the Broadcast Congested Clique

Abstract

In this paper, we bring the main tools of the Laplacian paradigm to the Broadcast Congested Clique. We introduce an algorithm to compute spectral sparsi ers in a polylogarithmic number of rounds, which directly leads to an e cient Laplacian solver. Based on this primitive, we consider the linear program solver of Lee and Sidford [LS14]. We show how to solve certain linear programs up to additive error 𝜖 with 𝑛 constraints on an 𝑛-vertex Broadcast Congested Clique network in O( √𝑛 log(1/𝜖)) rounds. Using this, we show how to nd an exact solution to the minimum cost ow problem in O( √ 𝑛) rounds.