The Laplacian Paradigm in Deterministic Congested Clique

04/05/2023

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by   Sebatian Forster, et al.

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In this paper, we bring the techniques of the Laplacian paradigm to the congested clique, while further restricting ourselves to deterministic algorithms. In particular, we show how to solve a Laplacian system up to precision ϵ in n^o(1)log(1/ϵ) rounds. We show how to leverage this result within existing interior point methods for solving flow problems. We obtain an m^3/7+o(1)U^1/7 round algorithm for maximum flow on a weighted directed graph with maximum weight U, and we obtain an Õ(m^3/7(n^0.158+n^o(1)polylog W)) round algorithm for unit capacity minimum cost flow on a directed graph with maximum cost W. Hereto, we give a novel routine for computing Eulerian orientations in O(log n log^* n) rounds, which we believe may be of separate interest.