A Distributed-Memory Algorithm for Computing a Heavy-Weight Perfect Matching on Bipartite Graphs

TL;DR

This paper introduces a scalable distributed-memory algorithm for finding high-weight perfect matchings in bipartite graphs, crucial for scalable sparse direct solvers, outperforming sequential methods in large-scale scenarios.

Contribution

It presents a fully parallel distributed algorithm that improves perfect matchings iteratively, enabling large-scale bipartite graph matching on supercomputers.

Findings

Scales efficiently up to 256 nodes (17,408 cores) on Cray XC40.

Produces near-optimal high-weight perfect matchings.

Handles problem instances too large for sequential algorithms.

Abstract

We design and implement an efficient parallel algorithm for finding a perfect matching in a weighted bipartite graph such that weights on the edges of the matching are large. This problem differs from the maximum weight matching problem, for which scalable approximation algorithms are known. It is primarily motivated by finding good pivots in scalable sparse direct solvers before factorization. Due to the lack of scalable alternatives, distributed solvers use sequential implementations of maximum weight perfect matching algorithms, such as those available in MC64. To overcome this limitation, we propose a fully parallel distributed memory algorithm that first generates a perfect matching and then iteratively improves the weight of the perfect matching by searching for weight-increasing cycles of length four in parallel. For most practical problems the weights of the perfect matchings generated by our algorithm are very close to the optimum. An efficient implementation of the algorithm scales up to 256 nodes (17,408 cores) on a Cray XC40 supercomputer and can solve instances that are too large to be handled by a single node using the sequential algorithm.