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

Wästlund, Johan

doi:10.1002/rsa.20066

Cited by 16 publications

(13 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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 . This conjecture has been subsequently established in [24] by Wästlund using a modification of the argument in [14].…”

Section: Background and Related Workmentioning

confidence: 90%

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

Nair

Prabhakar

Sharma

2005

Random Struct Algorithms

View full text Add to dashboard Cite

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 $\sum_{i=1}^n (1/i^2)$. 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. 2005

show abstract

Section: Background and Related Workmentioning

confidence: 90%

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

Nair

Prabhakar

Sharma

2005

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…Generalizing (9) to the case of matrix C comprised of either zeros or independent random variables distributed exponentially with parame- Wästlund (2005a) has validated the BuckChan-Robbins conjecture (8) as well. 4…”

Section: Minimum K-assignment and Coppersmith-sorkin And Buck-chan-romentioning

confidence: 91%

Random assignment problems

Krokhmal

Pardalos

2009

European Journal of Operational Research

View full text Add to dashboard Cite

“…Equation (46) is a generalization of the formula conjectured by Buck, Chan and Robbins [14] and proved in [47]. It can be established by following the same route as the proof of Theorem 7.3 in §7, but we shall give a different proof, generalizing to higher moments.…”

Section: Interpretation In Terms Of the Urn Processmentioning

confidence: 96%