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

Nair, Chandra; Prabhakar, Balaji; Sharma, Mayank

doi:10.1109/sfcs.2003.1238191

Cited by 19 publications

(22 citation statements)

References 20 publications

(28 reference statements)

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“…However, calculation of explicit constants in dimensions d Ն 2 seems beyond the reach of analytic techniques. For the mean-field bipartite MM problem, impressive recent work (26,27) has proven an exact formula giving the expectation of the finite-n minimum total matching length, though such exact methods seem unlikely to be widely feasible.…”

Section: Discussionmentioning

confidence: 99%

Scaling and universality in continuous length combinatorial optimization

Aldous

Percus²

2003

Proc. Natl. Acad. Sci. U.S.A.

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We consider combinatorial optimization problems defined over random ensembles and study how solution cost increases when the optimal solution undergoes a small perturbation ␦. For the minimum spanning tree, the increase in cost scales as ␦ 2 . For the minimum matching and traveling salesman problems in dimension d > 2, the increase scales as ␦ 3 ; this is observed in Monte Carlo simulations in d ‫؍‬ 2, 3, 4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems of this general kind into a small number of distinct categories, similar to universality classes in statistical physics.

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Section: Discussionmentioning

confidence: 99%

Scaling and universality in continuous length combinatorial optimization

Aldous

Percus²

2003

Proc. Natl. Acad. Sci. U.S.A.

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“…Let C n denote the cost of the minimum perfect matching in the n by n complete bipartite graph. The problem of establishing a good upper bound on var (C n ) for large n has been considered by several researchers [9], [32], [42]. The first proof that var (C n )!0 was obtained by M. Talagrand [42] with the method described in §9.…”

Section: The Variance Of Bipartite Matchingmentioning

confidence: 99%

The mean field traveling salesman and related problems

Wästlund

2010

Acta Math.

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“…So, if we replace e by f in T 1 we will, by (11), save at least 0.1/n. If we repeat this for all bad edges, then we will have a tree containing all of the Monday purchased edges and it will, in expectation, be at least 0.1 E[b] /n cheaper.…”

Section: Beyond the Threshold Heuristicmentioning

confidence: 99%

“…Another natural question might be to consider a 2-stage version of the random assignment problem. See Aldous [1,2], Linusson and Wästlund [10], and Nair, Prabhakar and Sharma [11] for recent work on the standard one-stage analysis. In principal, one could try to carry out a similar 2-stage probabilistic analysis for any combinatorial optimization problem.…”

Section: Open Questionsmentioning

confidence: 99%

On the random 2‐stage minimum spanning tree

Flaxman

Frieze

Krivelevich

2005

Random Struct Algorithms

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ABSTRACT:It is known [A. M. Frieze, Discrete Appl Math 10 (1985), [47][48][49][50][51][52][53][54][55][56] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to ζ(3) = ∞ i=1 i −3 . Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs c M : E → R and c T : E → R, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c M (e) and c T (e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M (e), or to wait until its Tuesday price c T (e) appears. The set of edges X M bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o(1). We show that, in the case of two-stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) − 1/2 + o(1). The threshold heuristic is shown to be sub-optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 − 1/e + o(1).

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

Abstract: Abstract

Cited by 19 publications

References 20 publications

Scaling and universality in continuous length combinatorial optimization

Scaling and universality in continuous length combinatorial optimization

The mean field traveling salesman and related problems

On the random 2‐stage minimum spanning tree

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