Scaling and universality in continuous length combinatorial optimization

Aldous, David; Percus, Allon G.

doi:10.1073/pnas.1635191100

Cited by 23 publications

(38 citation statements)

References 26 publications

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“…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

Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

Aldous¹,

Bordenave²,

Lelarge³

2009

SIAM J. Comput.

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A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over A ⊆ {1, 2, . . . , n}, the objective function |A| − i ξ i 1 1(i ∈ A, i + 1 ∈ A) for given ξ i > 0. This problem, with random (ξ i ), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion δ of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order δ 2 but not smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in AldousPercus (2003) of relating such scaling exponents to the algorithmic difficulty of optimization problems.

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“…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%

Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

Aldous¹,

Bordenave²,

Lelarge³

2009

SIAM J. Comput.

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“…(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%

Minimum spanning trees and random resistor networks inddimensions

Read¹

2005

Phys. Rev. E

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We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L, we show that there is a correction of order L θ , where θ ≤ 0 is a universal d-dependent exponent. There is a similar form for the change in optimum cost under a change in boundary condition. At non-zero temperature T , there is a crossover length ξ ∼ T −ν , such that on length scales larger than ξ, the behavior becomes that of uniform spanning trees. There is a scaling relation θ = −1/ν, and we provide several arguments that show that ν and −1/θ both equal νperc, the correlation length exponent for ordinary percolation in the same dimension d, in all dimensions d ≥ 1. The arguments all rely on the close relation of Kruskal's greedy algorithm for the minimum spanning tree, percolation, and (for some arguments) random resistor networks. The scaling of the entropy and free energy at small non-zero T , and hence of the number of near-optimal solutions, is also discussed. We suggest that the Steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions, as is the traveling salesman problem in two dimensions. Hence all will have the same value of θ = −3/4 in two dimensions. I. INTRODUCTIONMinimum spanning trees are a problem of combinatorial optimization [1,2]. Suppose we are given an undirected connected graph G, with vertex set V and edge set E, and a cost (or weight, or "length") ℓ ij assigned to each edge ij ∈ E (where i, j ∈ V ). The problem is to find a spanning tree T (i.e. a connected subgraph of G that includes all vertices in V , but whose edges form no cycles; such a tree must have |V | − 1 edges), such that the total cost of the edges in T,is as small as possible. Thus the minimization is over the set T of spanning trees in G.In this paper we are interested in the case in which G is a simply-connected portion Λ of a regular lattice in d ≥ 1 dimensions (with edges connecting nearest-neighbor lattice vertices only; the nearest-neighbor distance is fixed at 1 throughout this paper), including the case when Λ tends to the entire lattice, and the edge costs are independent, identically-distributed random variables, for example ℓ ij uniformly distributed on [0, 1]. We will also consider geometries with periodic boundary conditions, in which Λ has no boundary. The results also apply without significant modification to cases with other distributions, and/or with short-range correlations of the ℓ ij s, and to the Euclidean minimum spanning tree, in which N = |V | points are distributed independently and uniformly (with density 1) in a portion Λ of d-dimensional Euclidean space, and the cost of an edge ij is the Euclidean distance between i and j, for any pair i = j.The motivation for this work is to understand disordered systems at low temperatures better, beginning with those in which quantum-mechanical effects are negligible. Here "disordered" means that the Hamiltonian (or energy as a function of the system configuration) con...

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“…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%

Tracing the evolution of physics on the backbone of citation networks

2011

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Many innovations are inspired by past ideas in a non-trivial way. Tracing these origins and identifying scientific branches is crucial for research inspirations. In this paper, we use citation relations to identify the descendant chart, i.e. the family tree of research papers. Unlike other spanning trees which focus on cost or distance minimization, we make use of the nature of citations and identify the most important parent for each publication, leading to a tree-like backbone of the citation network. Measures are introduced to validate the backbone as the descendant chart. We show that citation backbones can well characterize the hierarchical and fractal structure of scientific development, and lead to accurate classification of fields and sub-fields.

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Scaling and universality in continuous length combinatorial optimization

Cited by 23 publications

References 26 publications

Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

Minimum spanning trees and random resistor networks inddimensions

Tracing the evolution of physics on the backbone of citation networks

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