ReFill: Reinforcement Learning for Fill-In Minimization

• URL: http://arxiv.org/abs/2501.16130v2
• Date: Thu, 30 Jan 2025 02:30:24 GMT
• Title: ReFill: Reinforcement Learning for Fill-In Minimization
• Authors: Elfarouk Harb, Ho Shan Lam,
• Abstract summary: Minimizing fill-in is NP-hard, and existing minimizations like Minimum Degree and Nested Dissection offer limited adaptability. We introduce textitReFill, a reinforcement learning framework enhanced by Graph Neural Networks (GNNs) to learn adaptive ordering strategies for fill-in.
• Score: 3.9134031118910264
• License:
• 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.

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