The Quick and the Dead: Estimating Sparse-Matrix Permanents with Adaptive Work Filtering

2026-08-15

AI Summary

This study addresses the challenges of premature trajectory termination and weight dispersion, which result in low effective sample sizes when estimating the permanent of sparse matrices. To overcome these limitations, we propose an adaptive work filtering mechanism that integrates minimum-degree ordering to achieve O(n+m) vertex selection complexity. Furthermore, an adaptive scheduling strategy is employed to execute computationally intensive operations, such as matrix scaling and filtering, on demand, thereby eliminating redundant calculations. Experimental results demonstrate that the proposed algorithm matches state-of-the-art performance on small-scale matrices while significantly improving estimation accuracy, effective sample size, and computational scalability for large-scale sparse matrices.

Abstract

Rasmussen's permanent estimator is a simple and unbiased estimator for the permanent of a binary matrix, but its practical performance can be limited by trajectories that terminate before completing a perfect matching. These failed trajectories, together with dispersion among the surviving weights, can substantially reduce the effective sample size. Although the literature leverages techniques such as matrix scaling to improve proposal balance and support filtering to remove structurally infeasible choices, using these at every step can substantially increase the trajectory cost. Furthermore, they do not directly address the choice of the next vertex. This paper uses the classical minimum-degree ordering in sparse matrix algorithms to select the next vertex with O(n + m) total bucket-maintenance work per trajectory, where n is the number of rows/columns in the matrix and m is the number of nonzeros. The proposed estimator uses adaptive schedules to invoke the more expensive scaling and filtering operations only when needed. The experiments show that it is competitive with the state of the art on the tested small matrices and scales effectively to large sparse matrices.

Problem

• Sparse-Matrix Permanents
• Rasmussen's permanent estimator
• Effective sample size
• Trajectory termination
• Adaptive work filtering

Innovation

• Adaptive Work Filtering
• Minimum-Degree Ordering
• Sparse-Matrix Permanents
• Rasmussen Estimator
• Adaptive Schedules