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ReFill: Reinforcement Learning for Fill-In Minimization

Elfarouk Harb, Ho Shan Lam

TL;DR

ReFill, a reinforcement learning framework enhanced by Graph Neural Networks to learn adaptive ordering strategies for fill-in minimization, is introduced, outperforming traditional heuristics by dynamically adapting to the structure of input matrices.

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.

ReFill: Reinforcement Learning for Fill-In Minimization

TL;DR

ReFill, a reinforcement learning framework enhanced by Graph Neural Networks to learn adaptive ordering strategies for fill-in minimization, is introduced, outperforming traditional heuristics by dynamically adapting to the structure of input matrices.

Abstract

Efficiently solving sparse linear systems , where 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.

Paper Structure

This paper contains 39 sections, 1 theorem, 8 equations, 21 figures, 2 tables.

Table of Contents

  1. Introduction and Background
  2. Minimum Degree Heuristic, MDH
  3. Minimum Fill-In Heuristic, MFillH
  4. Importance of Tie-Breaking.
  5. Nested Dissection
  6. How Do The Heuristics Compare?
  7. Motivation and Contributions
  8. A Bird's-Eye View of Our Method
  9. Reinforcement Learning for Combinatorial Optimization.
  10. Organization
  11. Related Work
  12. Evolution of Graph Neural Networks.
  13. Reinforcement Learning in Games.
  14. Proximal Policy Optimization (PPO).
  15. ReFill
  16. ...and 24 more sections

Key Result

Theorem 1.1

Given an ordering $\pi$ and assuming no "lucky cancellations"Lucky cancellations happens for some matrices ${\bf A}$ where some non-zero elements incidentally get cancelled to zeros when eliminating a variable. In practice, this is unlikely to happen, so one often ignores their effect., an edge $ij

Figures (21)

  • Figure 1: (a) A symmetric positive-definite matrix $\mathbf{A}$ and its factorization after Gaussian elimination with a fixed variable elimination order. This ordering introduces three fill-in entries at positions $A_{2,4}$, $A_{2,5}$, and $A_{4,5}$, which were initially zero but became non-zero during elimination. (b) The same matrix after applying a permutation $\pi = (5,1,2,3,4)$ via permutation matrix $\mathbf{P}_{\pi}$, which reorders the rows and columns of $\mathbf{A}$ by moving the first row to the last row, and the first column to the last position resulting in matrix $\mathbf{A}'$. This reordering leads to a matrix where Gaussian elimination introduces no fill-in, resulting in lower memory usage and faster computation.
  • Figure 2: (a) The matrix $\mathbf{A}$ from \ref{['fig:elimination_order_1']} and its corresponding graph representation $G(\mathbf{A})$. Vertices are eliminated in the order $1,2,3,4,5$. Eliminating vertex 1 forces its neighbors $(2,4,5)$ to form a clique, introducing three fill-in edges $(2,4), (2,5), (4,5)$ (shown in red) before vertex 1 is removed. These fill-in edges directly correspond to the non-zero entries added during Gaussian elimination in \ref{['fig:elimination_order_1']}. (b) The permuted matrix $\mathbf{A}'$ and its associated graph $G(\mathbf{A}')$, obtained by reordering the rows and columns of $\mathbf{A}$ by moving the first row to be the last row, and the first column to be the last column (i.e. according to the permutation $\pi=(51234)$). Although the graph is structurally identical to $G(\mathbf{A})$, the new vertex ordering eliminates all variables without introducing any fill-in edges, showing how proper ordering prevents unnecessary fill-in.
  • Figure 3: ReFill reinforcement learning environment. (a) An example observation $s$ is updated by deleting vertex 1, introducing 3 fill-in edges ($-3$ reward). (b) Schematic of the RL loop using a GNN and masked PPO.
  • Figure 4: Average fill-in (lower is better) on $8\times 8$ grid, comparing masking vs. no masking during training.
  • Figure 5: Average fill-in (lower is better) on $8\times 8$ grid, comparing masking vs. no masking during training.
  • ...and 16 more figures

Theorems & Definitions (1)

  • Theorem 1.1: rosetarjan