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

We present O(loglog n) -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+εapproximation of minimum vertex cover, for any n-vertex graph and any constant \eps>0. These improve the state of the art as follows: Our MIS algorithm leads to a simple O(loglog Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the Õ (√log Δ )-round algorithm of Ghaffari [PODC'17]. Our O(loglog n)-round (1+ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log^2log n)-round (1+\eps)-approximation algorithm of Czumaj et al. [STOC'18] and $O(loglog n)-round (1+ε)-approximation algorithm of Assadi et al. [arXiv'17]. Our O(loglog n)-round (2+ε)-approximate minimum vertex cover algorithm improves on an O(loglog n)-round O(1)-approximation of Assadi et al. [arXiv'17].