On the expected value of the minimum assignment

Abstract

The minimum k-assignment of an m by n matrix X is the minimum sum of k entries of X, no two of which belong to the same row or column. If X is generated by choosing each entry independently from the exponential distribution with mean 1, then Coppersmith and Sorkin conjectured that the expected value of its minimum k-assignment is \sum_{i,j \ge 0, i+j<k} 1/((m-i)(n-j)) and they (with Alm) have proven this for k < 5 and in certain cases when k=5 or k=6. They were motivated by the special case of k=m=n, where the expected value was conjectured by Parisi to be \sum_{i=1}^k 1/(i^2). In this paper we describe our efforts to prove the Coppersmith-Sorkin conjecture. We give evidence for the following stronger conjecture, which generalizes theirs. Conjecture. Suppose that r_1,...,r_m and c_1,...,c_n are positive real numbers. Let X be a random m by n matrix in which entry x_{ij} is chosen independently from the exponential distribution with mean 1/(r_ic_j). Then the expected value of the minimum k-assignment of X is \sum_{I,J} (-1)^{k - 1 - |I| - |J|} \binom{m + n - 1 - |I| - |J|}{k - 1 - |I| - |J|}\frac{1}{(\sum_{i \notin I}r_i) (\sum_{j \notin J} c_j)}. Here the sum is over proper subsets I of {1,...,m} and J of {1,...,n} whose cardinalities |I| and |J| satisfy |I|+|J|<k.