A Θ(m^9) ternary minimum-cost network flow LP model of the Assignment Problem polytope with applications to hard combinatorial optimization problems

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.