Exponentially Faster Massively Parallel Maximal Matching
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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+ε approximate maximum matching in O( n) rounds using 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 Ω( 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+Ω(1) per machine. We resolve this shortcoming by providing exponentially faster algorithms for maximal matching. Perhaps more importantly, we obtain this by analyzing an extremely simple and natural 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(Δ) 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 + ε approximate matching and 2 + ε approximate weighted matching. All these results can also be implemented in the congested clique model within O(Δ) rounds.
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