Random Assignment with Integer Costs

The random assignment problem is to minimize the cost of an assignment in a n×n matrix of random costs. In this paper we study this problem for some integer valued cost distributions. We consider both uniform distributions on 1, 2, . . . , m, for m = n or n, and random permutations of 1, 2, . . . , n for each row, or of 1, 2, . . . , n for the whole matrix. We find the limit of the expected cost for the n cases, and prove bounds for the n cases. This is done by simple coupling arguments together with Aldous recent results for the continuous case. We also present a simulation study of these cases.