Improved massively parallel computation algorithms for MIS, matching, and vertex cover

We present O(log log n)-round algorithms in the Massively Parallel Computation (MPC) model, with Õ(n) memory per machine, that compute a maximal independent set, a 1 + approximation of maximum matching, and a 2 + approximation of minimum vertex cover, for any n-vertex graph and any constant > 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, which improves on the Õ(log ∆)-round algorithm of Ghaffari [PODC'17]. • Our O(log log n)-round (1+)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log² log n)-round (1 +)-approximation algorithm of Czumaj et al. [STOC'18] and O(log log n)-round (1 + )- approximation algorithm of Assadi et al. [arXiv'17]. • Our O(log log n)-round (2 +)-approximate minimum vertex cover algorithm improves on an O(log log n)-round O(1)approximation of Assadi et al. [arXiv'17].