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In this paper, we present a new, network flow LP model of the standardAssignment Problem (AP) polytope. The model is not meant to be competitive withexisting standard procedures for solving the AP, as its complexity order ofsize is $O(m^9)$, where m is the number of assignments. However, it allows forhard combinatorial optimization problems (COPs) to be solved as AssignmentProblems (APs), including, in particular, the Quadratic, Cubic, Quartic,Quintic, and Sextic Assignment Problems, as well as the Traveling SalesmanProblem and many of its variations. Hence, in particular, the model re-affirms"P = NP." Illustrations are provided for the Linear Assignment (LAP), QuadraticAssignment (QAP), and Traveling Salesman (TSP) problems. Issues pertaining tothe extended formulations "barriers" for the LP modeling of hard COPs are notdiscussed in this paper because the developments are focused on the AssignmentProblem polytope only, and also the applicability/non-applicability of those"barriers" are thoroughly addressed in a separate paper* in which it is shownthat, in an optimization context, these "barriers" have no pertinence for amodel which projects to the AP polytope, provided appropriate costs can beattached to the non-superfluous variables of the model. Hence, the issues ofthe "barriers" are left out of this paper essentially for the sake of space.  *: Diaby, M., M. Karwan, and L. Sun [2024]. On modeling NP-Complete problemsas polynomial-sized linear programs: Escaping/Side-stepping the "barriers."Available at: arXiv:2304.07716 [cc.CC].

$O(m^9)$ network flow LP model of the Assignment Problem polytope with applications to hard combinatorial optimization problems