Exact Algorithms and Lower Bounds

Abstract

This chapter gives a summary on approaches and methods used for solving QAPs to optimality. Since the QAP is an NP-hard problem, only explicit (simple and straightforward) and implicit enumeration methods are known for solving it exactly. For a long time, branch and bound algorithms have been the most successful optimization approaches to QAPs, outperforming cutting plane algorithms whose convergence running time is simply unfeasible. Recently, theoretical results obtained on the combinatorial structure of the QAP polytope have raised new hopes that, in the future, polyhedral cutting planes can be successfully used for solving reasonably sized QAPs. Clearly, the design of efficient branch and cut methods in the vein of those already developed for the TSP (see for example [178]) is conditioned by the identification of new valid and possibly facet defining inequalities for the QAP polytope, and the development of the corresponding separation algorithms. Hence, quite a lot of efforts are required before the current size limits of solvable QAPs can be significantly improved.