Polyhedral Combinatorics of Quadratic Assignment Problems with Less Objects than Locations

Abstract

For the classical quadratic assignment problem (QAP) that requires n objects to be assigned to n locations (the n × n-case), polyhe- dral studies have been started in the very recent years by several authors. In this paper, we investigate the variant of the QAP, where the number of locations may exceed the number of objects (the m × n-case). It turns out that one can obtain structural results on the m × n-polytopes by exp- loiting knowledge on the n × n-case, since the first ones are certain pro- jections of the latter ones. Besides answering the basic questions for the affine hulls, the dimensions, and the trivial facets of the m × n-polytopes, we present a large class of facet defining inequalities. Employed into a cutting plane procedure, these polyhedral results enable us to compute optimal solutions for some hard instances from the QAPLIB for the first time without using branch-and-bound. Moreover, we can calculate for several yet unsolved instances significantly improved lower bounds.