Faster MPC Algorithms for Approximate Allocation in Uniformly Sparse Graphs

We study the allocation problem in the Massively Parallel Computation (MPC) model. This problem is a special case of $b$-matching, in which the input is a bipartite graph with capacities greater than $1$ in only one part of the bipartition. We give a $(1+\epsilon)$ approximate algorithm for the problem, which runs in $\tilde{O}(\sqrt{\log \lambda})$ MPC rounds, using sublinear space per machine and $\tilde{O}(\lambda n)$ total space, where $\lambda$ is the arboricity of the input graph. Our result is obtained by providing a new analysis of a LOCAL algorithm by Agrawal, Zadimoghaddam, and Mirrokni [ICML 2018], which improves its round complexity from $O(\log n)$ to $O(\log \lambda)$. Prior to our work, no $o(\log n)$ round algorithm for constant-approximate allocation was known in either LOCAL or sublinear space MPC models for graphs with low arboricity.