Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

Randomized composable coresets were introduced recently as an effective technique for solving matching and vertex cover problems in various models of computation. In this technique, one partitions the edges of a graph randomly into multiple pieces, compresses each piece into a smaller subgraph, namely a coreset, and solves the problem on the union of these coresets to find the solution. By designing small size randomized coresets, one can obtain efficient algorithms, in a black-box way, in multiple computational models including streaming, distributed communication, and the massively parallel computation (MPC) model. We develop randomized coresets of size $\widetilde{O}(n)$ that for any constant $\varepsilon > 0$, give a $(3/2+\varepsilon)$-approximation to matching and a $(3+\varepsilon)$-approximation to vertex cover. Our coresets improve upon the previously best approximation ratio of $O(1)$ for matching and $O(\log{n})$ for vertex cover. Our result for matching goes beyond a 2-approximation, which is a natural barrier for maximum matching in many models of computation. We further build on our coresets to achieve a $(1+\varepsilon)$-approximation to matching and $O(1)$-approximation to vertex cover in the MPC model with only $O(n/poly\log{(n)})$ memory per machine and $O(\log\log{(n)})$ MPC rounds. Our algorithm for vertex cover is the first $O(1)$-approximation MPC algorithm with $o(\log{n})$ rounds with $O(n)$ memory per machine and our matching algorithm improves upon the state-of-the-art $O((\log\log{n})^2)$ round algorithm of Czumaj et. al. (STOC 2018). A key technical ingredient of our paper is a novel application of edge degree constrained subgraphs (EDCS). At the heart of our proofs are new structural properties of EDCS that identify these subgraphs as sparse certificates for matchings and vertex covers which are quite robust to sampling and composition.