Congested Clique Algorithms for the Minimum Cut Problem

Abstract

We provide three different approaches to the minimum cut problem in the congested clique model of distributed computing. In this model, n nodes of the graph, each of which knows its own edges, can communicate in synchronous rounds; per round each node can send B-bits to each other node, where typically B=O(log n). At the end, each node should know its own part of the output, e.g., which side of the cut it is on. Our first algorithm is an O(1) round algorithm that finds a 1+o(1) approximation of the minimum cut. If the min-cut size is O(n^1/3 ), the algorithm finds an exact min-cut. This algorithm combines Karger's random sampling and his contraction algorithm; Nagamochi--Ibaraki--Nishizeki--Poljak's k--connectivity certificates; and Ahn--Guha--McGregor's algorithm for finding those certificates in the streaming model. To get an efficient implementation, we provide an algorithm that can solve simultaneously polynomially many instances of the MST problem in O(1) rounds. Our second algorithm is an O(log^3 n) round exact algorithm, based on the Karger-Stein approach. Its time complexity improves when larger messages are allowed. To implement this algorithm we present a general method to perform divide and conquer algorithms in the congested clique model. Our third algorithm is an O(log^2 n) round exact algorithm based on Karger's state of the art sequential exact min-cut algorithm, which works via tree-packing.