
Contents
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20.1 Belief propagation and the replica-symmetric analysis 20.1 Belief propagation and the replica-symmetric analysis
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20.1.1 The BP equations 20.1.1 The BP equations
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20.1.2 Statistical analysis 20.1.2 Statistical analysis
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20.1.3 BP-guided decimation 20.1.3 BP-guided decimation
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20.1.4 On the validity of the RS analysis 20.1.4 On the validity of the RS analysis
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20.2 Survey propagation and the 1RSB phase 20.2 Survey propagation and the 1RSB phase
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20.2.1 The SP(y)equations 20.2.1 The SP(y)equations
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20.2.2 The free entropy F RSB,e 20.2.2 The free entropy F RSB,e
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20.2.3 The large-y limit: The SP equations 20.2.3 The large-y limit: The SP equations
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20.2.4 The SAT–UNSAT threshold 20.2.4 The SAT–UNSAT threshold
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20.2.5 SP-guided decimation 20.2.5 SP-guided decimation
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20.3 Some ideas about the full phase diagram 20.3 Some ideas about the full phase diagram
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20.3.1 Entropy of clusters 20.3.1 Entropy of clusters
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20.3.2 The condensation transition for K ≥ 4 20.3.2 The condensation transition for K ≥ 4
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20.4 An exercise: Colouring random graphs 20.4 An exercise: Colouring random graphs
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Notes Notes
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20 Random K-satisfiability
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Published:January 2009
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Abstract
This chapter studies an ensemble of random satisfiability problems, ‘random K-satisfiability’ (K-SAT). Applying the 1RSB cavity method, it first derives the phase diagram in the limit of large N, in particular the location of the SAT-UNSAT threshold. Within the SAT phase, the chapter focuses on the intermediate clustered phase close, and computes the number of clusters to leading exponential order in N. The application of survey propagation to this problem is then described. Combined with a simple decimation procedure, the chapter provides an efficient method for finding satisfiable assignments in the clustered phase. The whole chapter is based on heuristic arguments. There is not yet any rigorous proof of the results presented, neither concerning the phase diagram, nor the convergence properties of message passing algorithms and their use in decimation procedures.
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