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 x n permutation matrices (identified with the symmetric group S) around its optimum,. (minimum or maximum). We estimate the fraction of permutations sigma such that f (sigma) lies within a given neighborhood of the optimal value of f and relate the optimal value with the average value of f over a neighborhood of the optimal permutation. We 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. Also, we identify a large class of functions f with the property that permutations close to the optimal permutation in the Hamming metric of S-n tend to produce near optimal values of f (such is, for example, the objective function in the symmetric Traveling Salesman Problem). We 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 S-n.