Combinatorics, Probability & Computing



Exact Expectations and Distributions for the Random Assignment Problem


SVEN ERICK ALM a1 and GREGORY B. SORKIN a2
a1 Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden (e-mail: sea@math.uu.se)
a2 Department of Mathematical Sciences, IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA (e-mail: sorkin@watson.ibm.com)

Abstract

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

Σi,j[greater-than-or-equal]0, i+j<k1/(mi)(nj).

Here, we prove the conjecture for k [less-than-or-eq, slant] 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 [rightward arrow] [infty infinity], the variance is 2/n + O(log n/n2). We also include some asymptotic properties of the expectation and variance when k is fixed.

(Received December 8 1999)
(Revised November 27 2001)