Linear programming insights into solvable cases of the quadratic assignment problem

The quadratic assignment problem is an NP-hard discrete optimization program that has been extensively studied for over 50 years. It has a variety of applications in many fields, but has proven itself extremely challenging to solve. As a result, an area of research has been to identify special cases which admit efficient solution strategies. This paper examines four such cases, and shows how each can be explained in terms of the dual region to the continuous relaxation of a classical linear reformulation of the problem known as the level-1 RLT representation. The explanations allow for simplifications and/or generalizations of the conditions defining the special cases.