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The replica symmetric phase of random constraint satisfaction problems

Part of: Equilibrium statistical mechanics Graph theory Theory of computing

Published online by Cambridge University Press:  03 December 2019

Amin Coja-Oghlan ,
Tobias Kapetanopoulos  and
Noela Müller
Amin Coja-Oghlan*
Affiliation:
Mathematics Institute, Goethe University, 10 Robert Mayer Str., Frankfurt60325, Germany.
Tobias Kapetanopoulos
Affiliation:
Mathematics Institute, Goethe University, 10 Robert Mayer Str., Frankfurt60325, Germany.
Noela Müller
Affiliation:
Mathematics Institute, Goethe University, 10 Robert Mayer Str., Frankfurt60325, Germany.
*
*Corresponding author. Email: [email protected]
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Abstarct

Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).

MSC classification

Primary: 05C80: Random graphs 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) 82B26: Phase transitions (general)
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 3 , May 2020 , pp. 346 - 422
Copyright
© Cambridge University Press 2019

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Footnotes

†

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement 278857–PTCC.

‡

Supported by a Stiftung Polytechnische Gesellschaft PhD grant.

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