We obtain a number of results regarding the distribution of values of a quadratic function f on the set of n × n permutation matrices (identified with the symmetric group Sn) around its optimum (minimum or maximum). In particular, we estimate the fraction of permutations σ such that f(σ) lies within a given neighborhood of the optimal value of f . We identify some “extreme” functions f (there a...

The quadratic assignment problem (QAP) was first proposed by Koopmans and Beckman [5] in the context of the plant location problem. Given n facilities, represented by the set F f1 fn , and n locations represented by the set L l1 ln , one must determine to which location each facility must be assigned. Let An n ai j be a matrix where ai j represents the flow between facilities fi and f j. Let Bn...

We study a generalization of the quadratic assignment problem (QAP) by allowing multiple equipments to be assigned at a single location as long as resources at the location permit. This problem arises in many real world applications such as facility location problem and logistics network design. We call the problem as the generalized quadratic assignment problem (GQAP) and show that this relaxa...

The quadratic assignment problem, QAP, is one of the most difficult of all combinatorial optimization problems. Here, we use an abbreviated application of the statistical mechanics replica method to study the asymptotic behavior of instances in which the entries of at least one of the two matrices that specify the problem are chosen from a random distribution P . Surprisingly, the QAP has not b...