Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with Õ(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2+ ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log∆)-round MIS algorithm in the Congested Clique model of distributed computing. This result improves exponentially on the Õ( √ log∆)-round algorithm of Ghaffari [PODC’17]. • Our O(log logn)-round (1 + ε)-approximate maximum matching algorithm simplifies and improves on a rather complex O(log logn)-round (1+ε)-approximation algorithm of Czumaj et al. [STOC’18]. • Our O(log logn)-round (2+ ε)-approximate minimum vertex cover algorithm improves on an O(log logn)-round O(1)-approximation of Assadi et al. [arXiv’17].