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

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2. Notations À cause des limitations du traitement de texte utilisé, les lettres N, Z, Q, R et C désigneront les ensembles usuels. Ces lettres pourront parfois avoir une autre signification, sans qu'une confusion soit possible. De même, nous utiliserons la lettre F pour désigner une partie finie de Z. La lettre P comme ensemble des indices d'une somme ou d'un produit désignera l'ensemble des nombres premiers. La notation [a, b], où a et b sont des entiers désignera l'ensemble {a, a + 1, ..., b}.

Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples of diffusers are defined by means of the action of elements of the matrix group SL(2, Z) on certain finite mathematical objects. Some corresponding, explicit expanders were introduced by Margulis. However, Gabber and Galil were the first to obtain good estimates for the expanders and produce from them a family of directed acyclic superconcentrators having density 271.8. In this paper we review various techniques for making expanders from diffusers. We also demonstrate asymptotic upper bounds on the strength of algebraically defined classes of degree k diffusers. Each upper bound is given as the norm of a diffusion operator on an infinite discrete group, and bounds for several examples are calculated. Numerical evidence is supplied in support of our conjecture that these bounds can be achieved by certain algebraically defined examples. The conjecture, if true, would lead to superconcentrators of density less than 58.

We prove two determinantal identities that generalize the Vandermonde determinant identity In the first of our identities the set {0, ..., m} indexing the rows and columns of the determinant is replaced by an arbitrary finite order ideal in the set of sequences of nonnegative integers which are 0 except for a finite number of components. In the second the index set is replaced by an arbitrary finite order ideal in the set of all partitions.