Stochastic Analysis of the Quadratic Assignment Problem

Let A and B be m × m matrices. The aim is to maximize Q (σ) = ∑ i, j a ij b σ( i )σ( j ) over all permutations σ of 1, 2, …, m . We define a simple “greedy” algorithm that constructs a permutation σ* that gives a large Q (σ*). Assuming that the entries of A are i.i.d. with a symmetric distribution, and that the entries of B are independent of A , we show under some weak moment conditions EQ (σ*) ≥ K −1 m 3/2 (log m ) 1/2 , where the constant k does not depend on m . This contrasts with the fact that max σ ≤ Km 3/2 (log m ) 1/2 with overwhelming probability.