A branch and bound algorithm for the quadratic assignment problem using a lower bound based on linear programming

In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. We report on computational experience with the branch and bound algorithm on all QAP test problems of dimension n less than or equal to 15 of QAPLIB, a standard Library of QAP test problems. The linear programming relaxations are solved with an implementation of an interior point algorithm that uses a preconditioned conjugate gradient algorithm to compute directions. The branch and bound algorithm is compared with a similar branch and bound algorithm that uses the Gilmore-Lawler lower bound (GLB) instead of the LP-based bound. The LP-based algorithm examines a small portion of the nodes explored by the GLB-based algorithm. Extensions to the implementation are discussed.