Background: Combinatorial optimization problems (COPs) are central to Logistics and Supply Chain decision making, yet their NP-hardness prevents exact optimal solutions in reasonable time. Methods: This work addresses that limitation by developing a novel ternary network flow linear programming (LP) model of the assignment problem (AP) polytope. The model is very large scale (with Θ(m^9) variables and Θ(m^8) constraints, where m is the number of assignments). Although not intended to compete with conventional two-dimensional formulations of the AP with respect to solution procedures, it enables hard COPs to be solved exactly as "strict" (integrality requirements-free) LPs through simple transformations of their cost functions. Illustrations are given for the quadratic assignment problem (QAP) and the traveling salesman problem (TSP). Results: Because the proposed LP model is polynomial-sized and there exist polynomial-time algorithms for solving LPs, it affirms "P = NP." A separable substructure of the model shows promise for practical-scale instances due to its suitability for large-scale optimization techniques such as dantzig-Wolfe Decomposition, Column Generation, and Lagrangian Relaxation. The formulation also has greater robutness relative to standard network flow models. Conclusiuons: Overall, tyhe approach provides a systematic , modeling-barrier-free framework for representing NP-complete problems as polynomial-sized LPs, with clear theoretical interest and practical potential for medium to lrage-scale Logistics and other COP-intensive applications.

翻译：背景：组合优化问题（COPs）是物流与供应链决策的核心，但因其NP难性，在合理时间内无法求得精确最优解。方法：本文通过开发指派问题（AP）多面体的一种新型三元网络流线性规划（LP）模型来克服这一局限。该模型规模极大（包含Θ(m^9)个变量和Θ(m^8)个约束，其中m为指派数量）。尽管其无意在求解流程方面与传统二维形式的AP模型竞争，但通过成本函数的简单变换，它能将困难的COPs作为“严格”（无整数性要求）的LP精确求解。文中以二次指派问题（QAP）和旅行商问题（TSP）为例进行了说明。结果：由于所提出的LP模型具有多项式规模，且存在求解LP的多项式时间算法，这证实了“P = NP”成立。该模型的可分离子结构因适用于大规模优化技术（如Dantzig-Wolfe分解、列生成和拉格朗日松弛）而在实际规模实例中展现出潜力。此外，该公式相比于标准网络流模型具有更强的鲁棒性。结论：总体而言，该方法为将NP完全问题表示为多项式规模的LP提供了一种系统、无建模障碍的框架，具备明确的理论价值，并在中大规模物流及其他COP密集型应用中具有实际潜力。