0 Ju n 20 00 A GENERALIZATION OF THE RANDOM ASSIGNMENT PROBLEM

We give a conjecture for the expected value of the optimal kassignment in an m × n-matrix, where the entries are all exp(1)-distributed random variables or zeros. We prove this conjecture in the case there is a zerocost k − 1-assignment. Assuming our conjecture, we determine some limits, as k = m = n → ∞, of the expected cost of an optimal n-assignment in an n by n matrix with zeros in some region. If we take the region outside a quarter-circle inscribed in the square matrix, this limit is thus conjectured to be π/24. We give a computer-generated verification of a conjecture of Parisi for k = m = n = 7 and of a conjecture of Coppersmith and Sorkin for k ≤ 5. We have used the same computer program to verify this conjecture also for k = 6.