Gromov-Wasserstein and optimal transport: from assignment problems to probabilistic numeric

2025-09-04

AI Summary

This work addresses structural matching between source and target objects residing in distinct metric-measure spaces—such as quadratic assignment problems (QAP) and graph matching—and unifies classical assignment problems within the optimal transport framework. We propose GW-MultiInit: a method grounded in the Gromov–Wasserstein (GW) distance, extended with Fused GW (FGW) to jointly align structural and feature information; it incorporates a multi-initialization strategy to mitigate local optima, combines entropy-regularized Sinkhorn solvers for efficiency, and integrates genetic algorithms for enhanced robustness. On capacity-constrained QAP benchmarks, GW-MultiInit efficiently yields near-optimal solutions and scales to large instances. Its parameterized variant enables flexible trade-offs between accuracy and computational efficiency. Overall, the approach significantly improves expressivity and practicality for cross-space structural alignment.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The assignment problem, a cornerstone of operations research, seeks an optimal one-to-one mapping between agents and tasks to minimize total cost. This work traces its evolution from classical formulations and algorithms to modern optimal transport (OT) theory, positioning the Quadratic Assignment Problem (QAP) and related structural matching tasks within this framework. We connect the linear assignment problem to Monge's transport problem, Kantorovich's relaxation, and Wasserstein distances, then extend to cases where source and target lie in different metric-measure spaces requiring Gromov-Wasserstein (GW) distances. GW formulations, including the fused GW variant that integrates structural and feature information, naturally address QAP-like problems by optimizing alignment based on both intra-domain distances and cross-domain attributes. Applications include graph matching, keypoint correspondence, and feature-based assignments. We present exact solvers, Genetic Algorithms (GA), and multiple GW variants, including a proposed multi-initialization strategy (GW-MultiInit) that mitigates the risk of getting stuck in local optima alongside entropic Sinkhorn-based approximations and fused GW. Computational experiments on capacitated QAP instances show that GW-MultiInit consistently achieves near-optimal solutions and scales efficiently to large problems where exact methods become impractical, while parameterized EGW and FGW variants provide flexible trade-offs between accuracy and runtime. Our findings provide theoretical foundations, computational insights, and practical guidelines for applying OT and GW methods to QAP and other real-world matching problems, such as those in machine learning and logistics.

Problem

• Extends optimal transport to solve quadratic assignment problems
• Develops Gromov-Wasserstein methods for cross-domain matching tasks
• Proposes scalable algorithms for large-scale structural matching problems

Innovation

• Gromov-Wasserstein distances for cross-domain alignment
• Multi-initialization strategy to avoid local optima
• Fused GW variant integrating structural and feature information