The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are linearizable; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form.

二次分配问题（QAP）是一个极具挑战性的 NP 难组合优化程序。由于其困难，研究重点是确定多项式可解的特殊情况。此强调中包括可线性化的实例；也就是说，它可以重写为线性分配问题，具有在所有可行解中保留目标函数值的属性。用于识别可线性化实例的各种已知的充分条件已根据 1 级重构线性化技术 (RLT) 形式的弱化版本的连续松弛进行了解释，该形式不会对变量子集强制执行非负性。此外，还给出了目标系数分解的充分必要条件。本文的主要贡献是确定了多面体理论和线性化之间的关系，为识别线性化实例提供了一种新颖但极其简单的必要和充分条件；具体来说，当且仅当相同弱化 RLT 形式的连续松弛有界时，QAP 的实例才是可线性化的。除了提供关于可线性化的 QAP 的新颖视角之外，这项研究的一个结果是，每个可线性化的实例都有一个二元 1 级 RLT 形式的（多项式大小）连续松弛的最优解。然而，反之则不然，因此连续松弛可以为不可线性化的 QAP 实例产生二元最优解。另一个结果是我们在提升的 RLT 变量空间中定义了一组最大线性无关方程；我们回答了最近的一个悬而未决的问题，即理论上最好的基于线性化的界限无法改进 1 级 RLT 形式。