Exponentially Faster Massively Parallel Maximal Matching

The study of graph problems in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Czumaj et al. [STOC'18], Assadi et al. [SODA'19], and Ghaffari et al. [PODC'18], gave algorithms for finding a $1+\varepsilon$ approximate maximum matching in $O(\log \log n)$ rounds using $\widetilde{O}(n)$ memory per machine. Despite this progress, we still have a far more limited understanding of the central symmetry-breaking problem of maximal matching. The round complexity of all these algorithms blows up to $\Omega(\log n)$ in this case, which is considered inefficient. In fact, the only known subpolylogarithmic round algorithm remains to be that of Lattanzi et al. [SPAA'11] which undesirably requires a strictly super-linear space of $n^{1+\Omega(1)}$ per machine. In this work, we close this gap by providing a novel analysis of an extremely simple and well-known algorithm. The algorithm edge-samples the graph, partitions the vertices at random, and finds a greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among some other results, leads to an $O(\log \log \Delta)$ round algorithm for maximal matching with $O(n)$ space. The space can be further improved to mildly sublinear in $n$ by standard techniques. As an immediate corollary, we get a $2$ approximation for minimum vertex cover in essentially the same rounds and space. This is the best possible approximation factor under standard assumptions, culminating a long line of research. Other corollaries include more efficient algorithms for $1 + \varepsilon$ approximate matching and $2 + \varepsilon$ approximate weighted matching. All these results can also be implemented in the congested clique model within $O(\log \log \Delta)$ rounds.