| Summary: | 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 + 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, which improves on the Õ(log ∆)-round algorithm of Ghaffari [PODC'17].
• Our O(log log n)-round (1+ ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log² log n)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC'18] and O(log log n)-round (1 + ε)- approximation algorithm of Assadi et al. [SODA'19].
• Our O(log log n)-round (2 + ε)-approximate minimum vertex cover algorithm improves on an O(log log n)-round O(1)approximation of Assadi et al. [arXiv'17].
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