  We give a conjecture for the expected value of the optimal k-assignment in an
m x n-matrix, where the entries are all exp(1)-distributed random variables or
zeros. We prove this conjecture in the case there is a zero-cost
$k-1$-assignment. Assuming our conjecture, we determine some limits, as
$k=m=n\to \infty$, of the expected cost of an optimal n -assignment in an n x
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 $\pi^2/24$. 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 $k\leq 5$.
We have used the same computer program to verify this conjecture also for k=6.
