Certain expected values in the random assignment problem

Lazarus, Andrew J.

doi:10.1016/0167-6377(93)90071-n

Cited by 26 publications

(36 citation statements)

References 2 publications

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“…Steele ([31] Chapter 4) goes on to give a detailed account of results of Walkup [35], Dyer et al [9], and Karp [14] which lead to the upper bound EA n ≤ 2. The lower bound lim sup n EA n ≥ 1 + e −1 was proved by Lazarus [18] and subsequent work of Olin [25] and Goemans and Kodialam [11] improved the lower bound to 1 51. Recently, Coppersmith and Sorkin [7] improved the upper bound to 1 94.…”

Section: Introductionmentioning

confidence: 83%

The ζ(2) limit in the random assignment problem

Aldous¹

2001

Random Struct Algorithms

249

417

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The random assignment (or bipartite matching) problem asks about A n = min π n i=1 c i π i , where c i j is a n × n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. used the replica method from statistical physics to argue nonrigorously that EA n → ζ 2 = π 2 /6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ 2 limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost-optimal matching coincides with the optimal matching except on a small proportion of edges.

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Section: Introductionmentioning

confidence: 83%

The ζ(2) limit in the random assignment problem

Aldous¹

2001

Random Struct Algorithms

249

417

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“…We survey some of the work; more details can be found in [21,6]. Early work used feasible solutions to the dual linear programming (LP) formulation of the assignment problem for obtaining the following lower bounds for C * : (1+1/e) by Lazarus [12], 1.441 by Goemans and Kodialam [8], and 1.51 by Olin [18]. The first upper bound of 3 was given by Walkup [23], who thus demonstrated that lim sup n E(C n ) is finite.…”

Section: Background and Related Workmentioning

confidence: 99%

“…Observe that one takes the edge belonging to S k 1 when going from a right vertex to a left vertex and a S k 3 -edge when going from a left vertex to a right vertex. Let v be the first vertex along this path that also belongs to P 12 .…”

Section: Casementioning

confidence: 99%

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

Nair

Prabhakar

Sharma

2005

Random Struct Algorithms

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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

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“…The proofs for both bounds are non-constructive and do not lead to heuristics which produce assignments with expected optimal value within the given bounds. Lower bounds for the expected optimal value of the LSAP with independent and uniformly distributed costs c ij on [0, 1] are given by Lazarus [120]. The author exploits the weak duality and evaluates the expected value of the dual objective function i u i + j v j achieved after the first iteration of the Hungarian method (row and column reductions, see also Section 3.1).…”

Section: Asymptotic Behavior and Probabilistic Analysismentioning

confidence: 99%

Linear Assignment Problems and Extensions

Burkard

Çela

1999

Handbook of Combinatorial Optimization

252

150

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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.

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Certain expected values in the random assignment problem

Cited by 26 publications

References 2 publications

The ζ(2) limit in the random assignment problem

The ζ(2) limit in the random assignment problem

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

Linear Assignment Problems and Extensions

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