On the expected value of the minimum assignment

Abstract

The minimum k-assignment of an m × 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 i,j≥0, i+j<kand they (with Alm) have proven this for k ≤ 4 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 k i=1 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 × n matrix in which entry x ij is chosen independently from the exponential distribution with mean 1 r i c j . Then the expected value of the minimum k-assignment of X isHere the sum is over proper subsets I of {1, . . . , m} and J of {1, . . . , n} whose cardinalities |I| and |J| satisfy |I| + |J| < k. * All three authors are affiliated with the Center for Communications Research, Princeton, NJ 08540.

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

“…Indeed, it is by proving this conjecture in their recent work [14] that they obtain proofs of the Parisi and Coppersmith-Sorkin conjectures. Buck, Chan, and Robbins [5] generalized the Coppersmith-Sorkin conjecture to the case where the c ij are distributed according to exp(a i b j ) for a i , b j > 0. In other words, if we let a = [a i ] and b = [b j ] be column vectors, then the rate matrix for the costs is of rank 1 and is of the form ab T .…”

“…As mentioned in the introduction, Coppersmith and Sorkin [5] conjectured that the expected cost of the minimum k-assignment in an m × n rectangular matrix P of i.i.d. exp (1) entries is…”

A proof of a conjecture of Buck, Chan, and Robbins on the expected value of the minimum assignment

“…In this paper we show that by combining the methods and ideas of [11] and [4], we can prove a simultaneous generalization of the Buck-Chan-Robbins conjecture and the main theorem of [11]. At the same time, this provides a considerable simplification of the proof of (1) and (2).…”

Proofs of the Parisi and Coppersmith-Sorkin conjectures for the finite random assignment problem

“…5 We regret the cumbersome notation; but we must keep track of three indices: one for the number of rows in the matrix (of size ), one for the size of the matching, , and one for the rank of the matching, , among matchings of size .…”

“…Indeed, it is by establishing this conjecture in their recent work [15] that they obtain proofs of the Parisi and Coppersmith-Sorkin conjectures. Buck, Chan and Robbins [5] generalize the Coppersmith-Sorkin conjecture for matrices with for .…”