Characterizing linearizable QAPs by the level-1 reformulation-linearization technique

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.