Massively Parallel Algorithms for $b$-Matching

TL;DR

This paper introduces a highly efficient parallel algorithm for approximating maximum weighted $b$-matchings, achieving near-logarithmic round complexity with minimal memory, extending previous work on simpler matching problems.

Contribution

It presents the first massively parallel algorithm for $b$-matching with $O( ext{log log} ar{d})$ rounds, generalizing prior algorithms for standard matchings.

Findings

Achieves $O( ext{log log} ar{d})$ round complexity.

Uses near-linear memory per machine.

Extends previous algorithms to the $b$-matching problem.

Abstract

This paper presents an $O(\log\log \bar{d})$ round massively parallel algorithm for $1+\epsilon$ approximation of maximum weighted $b$-matchings, using near-linear memory per machine. Here $\bar{d}$ denotes the average degree in the graph and $\epsilon$ is an arbitrarily small positive constant. Recall that $b$-matching is the natural and well-studied generalization of the matching problem where different vertices are allowed to have multiple (and differing number of) incident edges in the matching. Concretely, each vertex $v$ is given a positive integer budget $b_v$ and it can have up to $b_v$ incident edges in the matching. Previously, there were known algorithms with round complexity $O(\log\log n)$, or $O(\log\log \Delta)$ where $\Delta$ denotes maximum degree, for $1+\epsilon$ approximation of weighted matching and for maximal matching [Czumaj et al., STOC'18, Ghaffari et al. PODC'18; Assadi et al. SODA'19; Behnezhad et al. FOCS'19; Gamlath et al. PODC'19], but these algorithms do not extend to the more general $b$-matching problem.