A generalization of the random assignment problem asks the expected cost of the minimum-cost matching of cardinality k in a complete bipartite graph Km,n, with independent random
edge weights. With weights drawn from the exponential distribution with intensity 1, the
answer has been conjectured to be
Here, we prove the conjecture for k [les ] 4, k = m = 5, and k = m = n = 6, using
a structured, automated proof technique that results in proofs with relatively few cases.
The method yields not only the minimum assignment cost's expectation but the Laplace
transform of its distribution as well. From the Laplace transform we compute the variance
in these cases, and conjecture that, with k = m = n
→ ∞, the variance is 2/n + O(log n/n2).
We also include some asymptotic properties of the expectation and variance when k is