Optimization problems and replica symmetry breaking in finite connectivity spin glasses

Monasson, Rémi

doi:10.1088/0305-4470/31/2/012

Cited by 152 publications

(223 citation statements)

References 23 publications

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“…The exact computation of the 1RSB free energy is a very difficult task for a finite connectivity model [18]. Good results can be obtained the following variational Ansatz (see Ref.…”

Section: A2 Replica Symmetry Breakingmentioning

confidence: 99%

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Dynamic phase transition for decoding algorithms

et al. 2002

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The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, . . . ) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and from the dynamic points of view. We analyze the behavior of decoding algorithms using the mapping onto statistical-physics models. This allows to understand the intrinsic (i.e. algorithm independent) features of this behavior.

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“…The exact computation of the 1RSB free energy is a very difficult task for a finite connectivity model [18]. Good results can be obtained the following variational Ansatz (see Ref.…”

Section: A2 Replica Symmetry Breakingmentioning

confidence: 99%

“…The general one-step replica symmetry breaking order parameter [18] is In the previous Section we used a quasi-factorized Ansatz of the form: The meaning of this result is clear. For p > p d the system decompose in two parts.…”

Section: A21 Beyond the Factorized Ansatzmentioning

confidence: 99%

Dynamic phase transition for decoding algorithms

et al. 2002

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“…Such analyses involve order parameter functions, which generalize the replica matrices of [18]. The finite connectivity replica symmetry breaking theory (RSB) is still under development [5,19,20,21,6,22]. It appears that virtually all studies thus far have concentrated on equilibrium properties or microscopic approximation schemes [23], with the exception of a spherical model [24].…”

Section: Introductionmentioning

confidence: 99%

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

Hatchett¹,

Wemmenhove²,

Castillo³

et al. 2004

J. Phys. A: Math. Gen.

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Abstract. We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs our macroscopic laws close already in terms of the singlespin path probabilities at zero external field. For the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

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“…The replica method and cavity method are often used to estimate the free-energy density directly. Here, we use the replica method developed in an earlier study [30]. The alternative cavity method is presented in the Appendix, where the essentially same results derived in this section are obtained.…”

Section: B Lp-ip Modelmentioning

confidence: 99%

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems

Takabe

Hukushima

2016

Phys. Rev. E

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Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statisticalmechanical analyses reveal for α = 2 that the LP optimal solution is typically equal to that given by the IP below the critical average degree c = e in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result c = 1, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with α ≥ 3, minimum vertex covers on α-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree c = e/(α − 1) where the replica symmetry is broken.

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Optimization problems and replica symmetry breaking in finite connectivity spin glasses

Cited by 152 publications

References 23 publications

Dynamic phase transition for decoding algorithms

Dynamic phase transition for decoding algorithms

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems

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