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

In this paper, we present a new network flow linear programming (LP) model of the standard Assignment Problem (AP) polytope. The model is not meant to be competitive with the existing standard, two-dimensional abstraction of the AP with respect to solution procedures, as it is very-large-scale, with a variable space of dimension m^9, where m is the number of assignments. However, it allows for hard combinatorial optimization problems (COPs) to be solved as "strict" linear programs. Because the size complexity of the model is O(m^9), it affirms "P=NP." Conditions which can be used to assess the validity (or guide the formulations) of other models are developed. Illustrative applications to hard COPs are provided for the 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 because the developments in the paper are focused on the AP polytope only, and also because the applicability/non-applicability of those "barriers" in the context of the modeling framework used is thoroughly addressed in a separate paper*. Specific reasons why applications of the proposed modeling approach in variable spaces of dimension less than m^9 may not yield integral LP models are discussed (in an appendix), along with an illustrative numerical example. *: 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].