A Fourier space algorithm for solving quadratic assignment problems

The quadratic assignment problem (QAP) is a central problem in combinatorial optimization. Several famous computationally hard tasks, such as graph matching, partitioning, and the traveling salesman all reduce to special cases of the QAP. In this paper we propose a new approach to the QAP based on the theory of non–commutative Fourier analysis on the symmetric group. Specifically, we present a branch–and–bound algorithm that performs both the branching and the bounding steps in Fourier space. By exploiting the band–limited nature of the QAP objective function and using FFT techniques, the algorithm runs in O(n3) time per branch–and–bound node. The techniques underlying the algorithm generalize to a range of other combinatorial optimization problems.