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

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.