The Distribution of Values in the Quadratic Assignment Problem

Authors:
Alexander I. Barvinok;Tamon Stephen
Affiliations:
-;-
Venue:
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
Year:
2002

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Abstract

We obtain a number of results regarding the distribution of values of a quadratic function f on the set of n脳n permutation matrices (identified with the symmetric group Sn) around its optimum (minimum or maximum). We estimate the fraction of permutations s such that f(驴) 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, and show that for general f just the opposite behavior may take place.