ReFill: Reinforcement Learning for Fill-In Minimization

TL;DR

ReFill employs reinforcement learning with graph neural networks to learn adaptive elimination orderings, significantly reducing fill-in in sparse linear system solvers compared to traditional heuristics.

Contribution

The paper introduces ReFill, a novel reinforcement learning framework using GNNs to improve fill-in minimization strategies for sparse matrix factorizations.

Findings

ReFill outperforms traditional heuristics in reducing fill-in.

ReFill adapts effectively to diverse matrix structures.

Learning-based approach shows promise for classical sparse matrix problems.

Abstract

Efficiently solving sparse linear systems $Ax=b$, where $A$ is a large, sparse, symmetric positive semi-definite matrix, is a core challenge in scientific computing, machine learning, and optimization. A major bottleneck in Gaussian elimination for these systems is fill-in, the creation of non-zero entries that increase memory and computational cost. Minimizing fill-in is NP-hard, and existing heuristics like Minimum Degree and Nested Dissection offer limited adaptability across diverse problem instances. We introduce \textit{ReFill}, a reinforcement learning framework enhanced by Graph Neural Networks (GNNs) to learn adaptive ordering strategies for fill-in minimization. ReFill trains a GNN-based heuristic to predict efficient elimination orders, outperforming traditional heuristics by dynamically adapting to the structure of input matrices. Experiments demonstrate that ReFill outperforms strong heuristics in reducing fill-in, highlighting the untapped potential of learning-based methods for this well-studied classical problem.