Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures
نویسندگان
چکیده
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. When the cij are independent and identically distributed exponentials of mean 1, Parisi [Technical Report cond-mat/9801176, xxx LANL Archive, 1998] made the beautiful conjecture that the expected cost of the minimum assignment equals ∑n i=1(1/i 2). Coppersmith and Sorkin [Random Structures Algorithms 15 (1999), 113–144] generalized Parisi’s conjecture to the average value of the smallest k-assignment when there are n jobs and m machines. Building on the previous work of Sharma and Prabhakar [Proc 40th Annu Allerton Conf Communication Control and Computing, 2002, 657–666] and Nair [Proc 40th Annu Allerton Conf Communication Control and Computing, 2002, 667–673], we resolve the Parisi and Coppersmith-Sorkin conjectures. In the process we obtain a number of combinatorial results which may be of general interest. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 27, 413–444, 2005 Correspondence to: C. Nair *Supported by the Stanford Graduate Fellowship and Stanford Networking Research Center Grant 1005544-1WAAXI. †Supported in part by the NSF Grant ANI-9985446. ‡Supported by Stanford Office of Technology Grant 2DTA112, Stanford Networking Research Center Grant 1005545-1-WABCJ, and NSF Grant ANI-9985446. © 2005 Wiley Periodicals, Inc.
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