FRAM: Frobenius-Regularized Assignment Matching with Mixed-Precision Computing

2025-07-26

AI Summary

Graph matching is formulated as the Quadratic Assignment Problem (QAP), whose NP-hardness causes existing projection-based relaxation methods to suffer from excessive feasible set expansion, leading to numerical scale sensitivity and geometric inconsistency. To address this, we propose a novel Frobenius-regularized relaxation framework: a tunable Frobenius norm regularization term is introduced to suppress convex hull expansion, and the projection step is reformulated as a regularized linear assignment problem, solved via Scaling Doubly Stochastic Normalization (SDSN). We further design a mixed-precision solving architecture—leveraging CPU-GPU collaboration and adaptive FP16/FP32/FP64 arithmetic—to balance efficiency and accuracy. Theoretical analysis guarantees both geometric consistency and numerical robustness. Experiments demonstrate that our method outperforms state-of-the-art algorithms across all benchmarks on CPU; on GPU, it achieves up to 370× speedup with negligible solution accuracy degradation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Graph matching, typically formulated as a Quadratic Assignment Problem (QAP), seeks to establish node correspondences between two graphs. To address the NP-hardness of QAP, some existing methods adopt projection-based relaxations that embed the problem into the convex hull of the discrete domain. However, these relaxations inevitably enlarge the feasible set, introducing two sources of error: numerical scale sensitivity and geometric misalignment between the relaxed and original domains. To alleviate these errors, we propose a novel relaxation framework by reformulating the projection step as a Frobenius-regularized Linear Assignment (FRA) problem, where a tunable regularization term mitigates feasible region inflation. This formulation enables normalization-based operations to preserve numerical scale invariance without compromising accuracy. To efficiently solve FRA, we propose the Scaling Doubly Stochastic Normalization (SDSN) algorithm. Building on its favorable computational properties, we develop a theoretically grounded mixed-precision architecture to achieve substantial acceleration. Comprehensive CPU-based benchmarks demonstrate that FRAM consistently outperforms all baseline methods under identical precision settings. When combined with a GPU-based mixed-precision architecture, FRAM achieves up to 370X speedup over its CPU-FP64 counterpart, with negligible loss in solution accuracy.

Problem

• Addresses NP-hardness in graph matching via Frobenius-regularized relaxation
• Mitigates numerical scale sensitivity and geometric misalignment errors
• Enables efficient mixed-precision computing for accelerated performance

Innovation

• Frobenius-regularized Linear Assignment mitigates errors
• Scaling Doubly Stochastic Normalization algorithm efficiency
• Mixed-precision architecture enables substantial acceleration