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). We estimate the fraction of permutations σ such that f(σ) 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 Sn 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 Sn.
