2003
DOI: 10.1073/pnas.1635191100
|View full text |Cite
|
Sign up to set email alerts
|

Scaling and universality in continuous length combinatorial optimization

Abstract: 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… Show more

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1
1

Citation Types

1
37
0

Year Published

2004
2004
2012
2012

Publication Types

Select...
6
1
1

Relationship

0
8

Authors

Journals

citations
Cited by 23 publications
(38 citation statements)
references
References 26 publications
1
37
0
Order By: Relevance
“…The analog of Theorem 2 for mean-field-TSP, using Lagrange multipliers as in this paper, and leading to a non-rigorous argument that the scaling exponent equals 3, was given in [3].…”
Section: Parallels With the Cavity Methodsmentioning
confidence: 99%
See 1 more Smart Citation
“…The analog of Theorem 2 for mean-field-TSP, using Lagrange multipliers as in this paper, and leading to a non-rigorous argument that the scaling exponent equals 3, was given in [3].…”
Section: Parallels With the Cavity Methodsmentioning
confidence: 99%
“…A program initiated in [3] is to study this quantity for combinatorial optimization problems over random data. In this setting ε n (δ) becomes a random variable, but in many cases one expects that as n → ∞ there is a deterministic limit function ε(δ).…”
Section: Near-optimal Solutions In Combinatorial Optimizationmentioning
confidence: 99%
“…(A similar picture of TLSs for MSTs was also used in Ref. [12] to obtain the behavior of the cost of the minimum spanning tree that differs from the global MST by a given fraction of edges.) Note that in these statements we did not need to explicitly perform the disorder average, as the thermodynamic |V | → ∞ limit of these quantities self-averages.…”
Section: F Cost and Entropy At Positive Temperaturementioning
confidence: 99%
“…[12]). Correspondingly, ε , the change in the thermal (as well as ℓ ij ) average cost per vertex relative to the optimum, is…”
mentioning
confidence: 99%
“…The backbone hence results in a treelike structure and is found solely based on citation relations with no additional information. Similar concepts of spanning trees are extensively studied in transportation networks and oscillator networks, as minimum spanning trees in terms of traveling cost [9,10], and trees that maximize betweenness [11] or synchronizability [12]. Though the citation backbone can be constructed by these definitions, we see no direct correspondence between them and scientific descendant trees.…”
Section: Introductionmentioning
confidence: 95%