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

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3 Author(s)
Nair, C. ; Stanford Univ., CA, USA ; Prabhakar, B. ; Sharma, M.

Suppose that there are n jobs and n machines and it costs cij to execute job i on machine j. The assignment problem concerns the determination of a one-to-one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the classical random assignment problem has received a lot of interest in the recent literature, mainly due to the following pleasing conjecture of Parisi: The average value of the minimum-cost permutation in an n × n matrix with i.i.d. exp(1) entries equals Σi=1n 1/(i2). D. Coppersmith and G. Sorkin (1999) have generalized Parisi's conjecture to the average value of the smallest k-assignment when there are n jobs and m machines. We prove both conjectures based on a common set of combinatorial and probabilistic arguments.

Published in:

Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on

Date of Conference:

11-14 Oct. 2003