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

Abstract: In this paper, we present a new, network flow LP model of the standard Assignment Problem (AP) polytope. The model is not meant to be competitive with existing standard procedures for solving the AP, as its complexity order of size is $O(m^9)$, where m is the number of assignments. However, it allows for hard combinatorial optimization problems (COPs) to be solved as Assignment Problems (APs), including, in particular, the Quadratic, Cubic, Quartic, Quintic, and Sextic Assignment Problems, as well as the Traveling Salesman Problem and many of its variations. Hence, in particular, the model re-affirms "P = NP." Illustrations are provided for the Linear Assignment (LAP), Quadratic Assignment (QAP), and Traveling Salesman (TSP) problems. Issues pertaining to the extended formulations "barriers" for the LP modeling of hard COPs are not discussed in this paper because the developments are focused on the Assignment Problem polytope only, and also the applicability/non-applicability of those "barriers" are thoroughly addressed in a separate paper* in which it is shown that, in an optimization context, these "barriers" have no pertinence for a model which projects to the AP polytope, provided appropriate costs can be attached to the non-superfluous variables of the model. Hence, the issues of the "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 problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers." Available at: arXiv:2304.07716 [cc.CC].