Faster MPC Algorithms for Approximate Allocation in Uniformly Sparse Graphs

TL;DR

This paper introduces faster MPC algorithms for approximate allocation in sparse graphs, reducing round complexity from logarithmic to logarithmic in arboricity, enabling efficient solutions in distributed settings.

Contribution

It provides a new analysis of a LOCAL algorithm, achieving a $(1+ ext{epsilon})$ approximation with significantly fewer MPC rounds in low-arboricity graphs.

Findings

Achieves $(1+ ext{epsilon})$ approximation in $ ilde{O}( oot{ ext{log} ext{lambda}})$ rounds.

Reduces round complexity from $O( ext{log} n)$ to $O( ext{log} ext{lambda})$.

First $o( ext{log} n)$ round algorithm for low-arboricity graphs in MPC.

Abstract

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.