1. Biointelligence Laboratory, Seoul National University
Ch 2. SIMULATED ANNEALING 4.5 ~ 4.6 Adaptive Cooperative Systems, Martin Beckerman, 1997.
Summarized by
J.-W. Ha
Biointelligence Laboratory, Seoul National University

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
4.5 Spin Glasses
4.5.1 Introduction
4.5.2 Annealed and Quenched Random Variables
4.5.3 Replicas
4.5.4 The TAP Equations
4.5.5 The Parisi Order Parameter and Ultrametricity
4.5.6 Critical Behavior
4.5.7 The Energy Landscape and Ergodicity Breaking
4.6 Combinatorial Optimization and NP-Completeness
4.6.1 Combinatorial Optimization
4.6.2 The World of NP-Comlpleteness
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3. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Introduction
Spin Glass
Materials in which interactions between elements are random and conflicting
A low-temparature phase in which local magnetic moments are frozen into spatially random equilibrium orientations.
EA model (Edwards-Anderson)
SK model (Sherrington-Kirkpatrick)
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4. Annealed and Quenched Random Variables
J is quenched random variable
{J}-independent free energy
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5. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Replicas
Replica Trick
the pseudo-random interaction between different magnetic dipoles means that the orientation of these dipoles that has lowest energy is very difficult to predict, and is far richer than in an analogous ferromagnetic system.
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6. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
The TAP Equations
TAP mean-field equations
An alternative way of solving the SK model
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7. The Parisi Order Parameter and Ultrametricity
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8. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Critical Behavoir
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9. The Energy Landscape and Ergodicity Breaking
Ergodicity is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities. In an Ising ferromagnet below Tc this does not happen. If T < Tc, never mind how close they are, the system has chosen a global magnetization, and the phase space is divided into two regions. From one of them it is impossible to reach the other, unless a magnetic field is applied, or temperature is raised above Tc.
Finding the energy minima of 3D Ising spin glasses : NP-hard
2D case : P / but if external magnetic fields are present : NP-hard
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10. Combinatorial Optimization
Find the solution that is in some sense best from among a finite or countably infinite set of alternatives
TSP : traveling salesman problem
Get the permutation of cities with minimum length cycle
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11. The World of NP-Completeness
Hierarchical Classification Scheme on Problem Hardness
Hardness or complexity : time
P (polynomial) and NP (nondeterministic polynomial)
P ⊆NP but not proved P ≠ NP
NP problem
Intractable
Time complexity function cannot be bounded in polynomial time (exponential)
Two classes : existence (NP) and verification (P)
NP-Complete
A polynomial transformable set among NP
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