Learning Fill-in Reduction Ordering via Graph Policy Optimization for Sparse Matrices

2026-05-17

AI Summary

This work addresses the NP-complete problem of fill-in minimization in large-scale sparse matrix factorization, which incurs substantial memory and computational overhead. The authors propose a graph neural network–based matrix reordering method that uniquely integrates global multi-hop graph embeddings with local symbolic factorization feedback. To enhance training convergence and ordering quality, they introduce an adaptive saturation function for aligning policy and value networks. Experimental evaluation on the SuiteSparse matrix collection demonstrates that the proposed approach reduces fill-in by 29.3% and peak memory usage by 31.3% on average compared to the current state-of-the-art baseline.

Abstract

Matrix reordering in large sparse solvers seeks a permutation that minimizes factorization fill-in to reduce memory and computation. Because the minimum fill-in ordering problem is NP-complete and fill-in is implicit in the sparsity pattern, graph-theoretic heuristics are used. Existing reinforcement learning methods either ignore sparsity patterns--missing the global fill-in--or lack local exact fill-in feedback. We propose a graph policy optimization method, modeling fill-ins from global and local views: both the policy and value networks use a multi-hop graph neural backbone to embed global fill-in; the policy further interacts with symbolic factorization over graphs to extract local, step-level fill-ins, and the resulting feedback is aligned with the value network via an adaptive saturation function to improve convergence. On the SuiteSparse Matrix Collection, our method achieves mean reductions of 29.3 in fill-ins and 31.3 in peak memory usage over state-of-the-art baselines.

Problem

• fill-in reduction
• sparse matrices
• matrix reordering
• graph policy optimization
• symbolic factorization

Innovation

• graph policy optimization
• fill-in reduction
• sparse matrix reordering
• symbolic factorization
• graph neural networks