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

arXiv: https://arxiv.org/abs/2605.17362 · HTML: https://mlsanity.com/paper/2605.17362/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 22 references that are in the map.
- First posted on arXiv 17 May 2026.

## Builds on (most cited first, 6 of 6)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Graph Partitioning and Sparse Matrix Ordering using Reinforcement Learning and Graph Neural Networks](https://mlsanity.com/paper/2104.03546/) (2021)
- [Alpha Elimination: Using Deep Reinforcement Learning to Reduce Fill-In during Sparse Matrix Decomposition](https://mlsanity.com/paper/2310.09852/) (2023)
- [Collective Influence Algorithm to find influencers via optimal percolation in massively large social media](https://mlsanity.com/paper/1603.08273/) (2016)

## Abstract (arXiv)

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.