The Distribution of Values in the Quadratic Assignment Problem

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 are 4 of those for n even and 5 for n odd) such that the distribution of every quadratic function around its optimum is a certain “mixture” of the distributions of the extremes and describe a natural class of functions (which includes, for example, the objective function in the Traveling Salesman Problem) with a relative abundance of near-optimal permutations. In particular, we identify a large class of functions f with the property that permutations in the vicinity of the optimal permutation (in the Hamming metric of Sn) tend to produce near optimal values of f (such is, for example, the objective function in the symmetric Traveling Salesman Problem) and show that for general f , just the opposite behavior may take place: an average permutation in the vicinity of the optimal permutation may be much worse than an average permutation in the whole group Sn.