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

from arxiv, Current Version (#11): Major rewrite with improved formalization. Why lower-dimensional may not yield integral LP models is discussed, along with an illustrative numerical example

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].

翻译：本文提出了一种用于标准指派问题多面体的新型网络流线性规划模型。该模型在求解过程方面并非旨在与现有的标准二维抽象模型竞争，因为它规模极大，变量空间维度为m^9，其中m为指派数量。然而，它允许将困难的组合优化问题作为“严格”线性规划来求解。由于模型的规模复杂度为O(m^9)，这证实了“P=NP”。本文还提出了可用于评估其他模型有效性（或指导其构建）的条件。针对二次指派问题和旅行商问题，提供了在困难组合优化问题中的示例应用。本文未讨论关于困难组合优化问题线性规划建模中扩展公式“障碍”的相关问题，因为论文的进展仅聚焦于指派问题多面体，并且这些“障碍”在所采用建模框架背景下的适用性/非适用性已在另一篇独立论文*中得到彻底阐述。文中（在附录中）讨论了所提出的建模方法在维度小于m^9的变量空间中应用可能无法产生整数线性规划模型的具体原因，并附有说明性数值示例。*：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].