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 $\tilde{O}(n)$ memory per machine, that compute a maximal independent set, a $1+\epsilon$ approximation of maximum matching, and a $2+\epsilon$ approximation of minimum vertex cover, for any $n$-vertex graph and any constant $\epsilon>0$. These improve the state of the art as follows: -- Our MIS algorithm leads to a simple $O(\log\log \Delta)$-round MIS algorithm in the Congested Clique model of distributed computing. This result improves exponentially on the $\tilde{O}(\sqrt{\log \Delta})$-round algorithm of Ghaffari [PODC'17]. -- Our $O(\log\log n)$-round $(1+\epsilon)$-approximate maximum matching algorithm simplifies and improves on a rather complex $O(\log^2\log n)$-round $(1+\epsilon)$-approximation algorithm of Czumaj et al. [STOC'18]. -- Our $O(\log\log n)$-round $(2+\epsilon)$-approximate minimum vertex cover algorithm improves on an $O(\log\log n)$-round $O(1)$-approximation of Assadi et al. [arXiv'17].