# A Parallel Solver for Graph Laplacians

## TL;DR

This paper introduces a parallel multigrid solver for graph Laplacians that efficiently handles large, irregular graphs in distributed memory environments, significantly outperforming existing methods.

## Contribution

It presents a novel parallel aggregation and low degree elimination algorithms tailored for irregular graphs, enabling scalable solutions for massive graph Laplacian problems.

## Key findings

- Scales to 576 processes and 1.7 billion edges
- Outperforms existing parallel solvers in speed and efficiency
- Effective for irregular degree graphs in distributed settings

## Abstract

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.