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The replica symmetric phase of random constraint satisfaction problems

Part of: Theory of computing Equilibrium statistical mechanics Graph theory

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.

References

Abbe, E. (2017) Community detection and stochastic block models: Recent developments. J. Mach. Learn. Res. 18 6446–6531.Google Scholar
Abbe, E. and Sandon, C. (2018) Proof of the achievability conjectures for the general stochastic block model. CPAM 71 1334–1406.Google Scholar
Achlioptas, D. and Coja-Oghlan, A. (2008) Algorithmic barriers from phase transitions. In Proc. 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 793–802.CrossRefGoogle Scholar
Achlioptas, D., Chtcherba, A., Istrate, G. and Moore, C. (2001) The phase transition in 1-in-k SAT and NAE 3-SAT. In Proc. 12th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 721–722.Google Scholar
Achlioptas, D. and Moore, C. (2006) Random k-SAT: Two moments suffice to cross a sharp threshold. SIAM J. Comput. 36 740–762.CrossRefGoogle Scholar
Achlioptas, D. and Naor, A. (2005) The two possible values of the chromatic number of a random graph. Ann. of Math. 162 1333–1349.CrossRefGoogle Scholar
Achlioptas, D., Naor, A. and Peres, Y. (2005) Rigorous location of phase transitions in hard optimization problems. Nature 435 759–764.CrossRefGoogle ScholarPubMed
Achlioptas, D. and Peres, Y. (2004) The threshold for random k-SAT is 2k ln 2 – O(k). J. Amer. Math. Soc. 17 947–973.CrossRefGoogle Scholar
Alon, N. and Kahale, N. (1997) A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26 1733–1748.CrossRefGoogle Scholar
Anastos, M., Frieze, A. and Pegden, W. (2018) Constraining the clustering transition for colorings of sparse random graphs. Electron. J. Combin. 25 P1.72CrossRefGoogle Scholar
Applebaum, B. and Lovett, S. (2018) Algebraic attacks against random local functions and their countermeasures. SIAM J. Comput. 47 52–79.CrossRefGoogle Scholar
Ayre, P., Coja-Oghlan, A., Gao, P. and Müller, N (2019) The satisfiability threshold for random linear equations Combinatorica, in press.CrossRefGoogle Scholar
Ayre, P., Coja-Oghlan, A. and Greenhill, C. (2019) Hypergraph coloring up to condensation. Random Struct. Alg. 54 615–652.CrossRefGoogle Scholar
Ayre, P. and Greenhill, C. (2019) Rigid colorings of hypergraphs and contiguity. SIAM J. Discrete Math. 33 1575–1606.CrossRefGoogle Scholar
Bandyopadhyay, A. and Gamarnik, D. (2008) Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Struct. Alg. 33 452–479.CrossRefGoogle Scholar
Banks, J., Moore, C., Neeman, J. and Netrapalli, P. (2016) Information-theoretic thresholds for community detection in sparse networks. In Proc. 29th Annual Conference on Learning Theory (COLT), pp. 383–416.Google Scholar
Bapst, V. and Coja-Oghlan, A. (2016) Harnessing the Bethe free energy. Random Struct. Alg. 49 694–741.CrossRefGoogle ScholarPubMed
Bapst, V., Coja-Oghlan, A. and Efthymiou, C. (2017) Planting colourings silently. Combin. Probab. Comput. 26 338–366.CrossRefGoogle Scholar
Bapst, V., Coja-Oghlan, A., Hetterich, S., Rassmann, F. and Vilenchik, D. (2016) The condensation phase transition in random graph coloring. Comm. Math. Phys. 341 543–606.CrossRefGoogle Scholar
Bayati, M., Gamarnik, D. and Tetali, P. (2013) Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 4080–4115.CrossRefGoogle Scholar
Bollobás, B. (2001) Random Graphs, second edition. Cambridge University Press.CrossRefGoogle Scholar
Cheeseman, P., Kanefsky, B. and Taylor, W. (1991) Where the really hard problems are. In Proc. 12th International Joint Conference on Artificial Intelligence (IJCAI), pp. 331–337.Google Scholar
Chvátal, V. and Reed, B. (1992) Mick gets some (the odds are on his side). In Proc. 33rd Annual Symposium on Foundations of Computer Science (FOCS), IEEE, pp. 620–627.CrossRefGoogle Scholar
Coja-Oghlan, A. (2013) Upper-bounding the k-colorability threshold by counting covers. Electron. J. Combin. 20 P32.CrossRefGoogle Scholar
Coja-Oghlan, A., Efthymiou, C., Jaafari, N., Kang, M. and Kapetanopoulos, T. (2018) Charting the replica symmetric phase. Comm. Math. Phys. 359 603–698.CrossRefGoogle Scholar
Coja-Oghlan, A. and Jaafari, N. (2016) On the Potts model on random graphs. Electron. J. Combin. 23 P4.3.CrossRefGoogle Scholar
Coja-Oghlan, A., Krzakala, F., Perkins, W. and Zdeborova, L. (2018) Information-theoretic thresholds from the cavity method. Adv. Math. 333 694–795.CrossRefGoogle Scholar
Coja-Oghlan, A. and Panagiotou, K. (2012) Catching the k-NAESAT threshold. In Proc. 44th Annual ACM Symposium on Theory of Computing (STOC), pp. 899–908.CrossRefGoogle Scholar
Coja-Oghlan, A. and Panagiotou, K. (2016) The asymptotic k-SAT threshold. Adv. Math. 288 985–1068.CrossRefGoogle Scholar
Coja-Oghlan, A. and Reichman, D. (2013) Sharp thresholds and the partition function. J. Statist. Phys. Conf. Ser. 473 012015.CrossRefGoogle Scholar
Coja-Oghlan, A. and Wormald, N. (2018) The number of satisfying assignments of random regular k-SAT formulas. Combin. Probab. Comput. 4 496–530.CrossRefGoogle Scholar
Coja-Oghlan, A. and Zdeborová, L. (2012) The condensation transition in random hypergraph 2-coloring. In Proc. 23rd Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 241–250.CrossRefGoogle Scholar
Decelle, A., Krzakala, F., Moore, C. and Zdeborová, L. (2011) Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84 066106.CrossRefGoogle ScholarPubMed
Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M. (2010) Tight thresholds for cuckoo hashing via XORSAT. International Colloquium on Automata, Languages, and Programming, Springer, pp. 213–225.CrossRefGoogle Scholar
Ding, J., Sly, A. and Sun, N. (2015) Proof of the satisfiability conjecture for large k. In Proc. 47th Annual ACM Symposium on Theory of Computing (STOC), pp. 59–68.CrossRefGoogle Scholar
Ding, J., Sly, A. and Sun, N. (2016) Satisfiability threshold for random regular NAE-SAT. Comm. Math. Phys. 341 435–489.CrossRefGoogle Scholar
Dubois, O. and Mandler, J. (2002) The 3-XORSAT threshold. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 769–778.CrossRefGoogle Scholar
Dyer, M. and Frieze, A. (1989) The solution of some NP-hard problems in polynomial expected time. J. Algorithms 10 451–489.CrossRefGoogle Scholar
Dyer, M., Frieze, A. and Greenhill, C. (2015) On the chromatic number of a random hypergraph. J. Combin. Theory Ser. B 113 68–122.CrossRefGoogle Scholar
Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 5 17–61.Google Scholar
Feige, U. (2002) Relations between average case complexity and approximation complexity. In Proc. 24th Annual ACM Symposium on Theory of Computing (STOC), pp. 534–543.CrossRefGoogle Scholar
Feldman, V., Perkins, W. and Vempala, S. (2015) On the complexity of random satisfiability problems with planted solutions. In Proc. 48th Annual ACM Symposium on Theory of Computing (STOC), pp. 77–86.CrossRefGoogle Scholar
Ferrari, U., Lucibello, C., Morone, F., Parisi, G., Ricci-Tersenghi, F. and Rizzo, T. (2013) Finite-size corrections to disordered systems on Erdőos–Rényi random graphs. Phys. Rev. B 88 184201.CrossRefGoogle Scholar
Franz, S. and Leone, M. (2003) Replica bounds for optimization problems and diluted spin systems. J. Statist. Phys. 111 535–564.CrossRefGoogle Scholar
Galanis, A., Stefankovic, D. and Vigoda, E. (2015) Inapproximability for antiferromagnetic spin systems in the tree nonuniqueness region. J. Assoc. Comput. Mach. 62 50.CrossRefGoogle Scholar
Gamarnik, D. and Sudan, M. (2014) Limits of local algorithms over sparse random graphs. In Proc. 5th Conference on Innovations in Theoretical Computer Science (ITCS), pp. 369–376.CrossRefGoogle Scholar
Gamarnik, D. and Sudan, M. (2017) Performance of sequential local algorithms for the random NAE-K-SAT problem. SIAM J. Comput. 46 590–619.CrossRefGoogle Scholar
Gerschenfeld, A. and Montanari, A. (2007) Reconstruction for models on random graphs. In Proc. 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 194–204.CrossRefGoogle Scholar
Goerdt, A. (1996) A threshold for unsatisfiability. J. Comput. Syst. Sci. 53 469–486CrossRefGoogle Scholar
Goldreich, O. (2000) Candidate one-way functions based on expander graphs. Cryptology ePrint Archive, report 2000/063. http://eprint.iacr.org/2000/063Google Scholar
Graf, S. and Luschgy, H. (2007) Foundations of Quantization for Probability Distributions, Springer.Google Scholar
Graf, S. and Luschgy, H. (2009) Quantization for probability measures in the Prokhorov metric. SIAM Theory Probab. Appl. 53 216–241.CrossRefGoogle Scholar
Guerra, F. (2003) Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.CrossRefGoogle Scholar
van der Hofstad, R. (2017) Random Graphs and Complex Networks, Cambridge University Press.CrossRefGoogle Scholar
Janson, S. (1995) Random regular graphs: Asymptotic distributions and contiguity. Combin. Probab. Comput. 4 369–405.CrossRefGoogle Scholar
Kesten, H. and Stigum, B. (1966) Additional limit theorem for indecomposable multidimensional Galton–Watson processes. Ann. Math. Statist. 37 1463–1481.CrossRefGoogle Scholar
Krivelevich, M. and Vilenchik, D. (2006) Semirandom models as benchmarks for coloring algorithms. In Proc. 3rd Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 211–221.CrossRefGoogle Scholar
Krzakala, F., Mézard, M. and Zdeborová, L. (2014) Reweighted belief propagation and quiet planting for random K-SAT. J. Satisfiability, Boolean Modeling and Computation 8 149–171.CrossRefGoogle Scholar
Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G. and Zdeborová, L. (2007) Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Nat. Acad. Sci. 104 10318–10323.Google Scholar
Krzakala, F. and Zdeborová, L. (2009) Hiding quiet solutions in random constraint satisfaction problems. Phys. Rev. Lett. 102 238701.CrossRefGoogle ScholarPubMed
Mézard, M. and Montanari, A. (2009) Information, Physics and Computation, Oxford University Press.CrossRefGoogle Scholar
Mézard, M., Parisi, G. and Zecchina, R. (2002) Analytic and algorithmic solution of random satisfiability problems. Science 297 812–815.CrossRefGoogle ScholarPubMed
Molloy, M. (2012) The freezing threshold for k-colourings of a random graph. In Proc. 43rd Annual ACM Symposium on Theory of Computing (STOC), pp. 921–930.CrossRefGoogle Scholar
Montanari, A., Restrepo, R. and Tetali, P. (2011) Reconstruction and clustering in random constraint satisfaction problems. SIAM J. Discrete Math. 25 771–808.CrossRefGoogle Scholar
Moore, C. (2017) The computer science and physics of community detection: Landscapes, phase transitions, and hardness. Bulletin EATCS 121.Google Scholar
Mossel, E., Neeman, J. and Sly, A. (2015) Reconstruction and estimation in the planted partition model. Probab. Theory Rel. Fields 162 431–461.CrossRefGoogle Scholar
Panchenko, D. and Talagrand, M. (2004) Bounds for diluted mean-fields spin glass models. Probab. Theory Rel. Fields 130 319–336.CrossRefGoogle Scholar
Pittel, B. and Sorkin, G. (2016) The satisfiability threshold for k-XORSAT. Combin. Probab. Comput. 25 236–268.CrossRefGoogle Scholar
Rassmann, F. (2019) On the number of solutions in random graph k-colouring. Combin. Probab. Comput. 28 130–158.CrossRefGoogle Scholar
Robinson, R. and Wormald, N. (1992) Almost all cubic graphs are Hamiltonian. Random Struct. Alg. 3 117–125.CrossRefGoogle Scholar
Schaefer, T. (1978) The complexity of satisfiability problems. In Proc. 10th Annual ACM Symposium on Theory of Computing (STOC), pp. 216–226.CrossRefGoogle Scholar
Shamir, E. and Spencer, J. (1987) Sharp concentration of the chromatic number of random graphs G n,p. Combinatorica 7 121–129CrossRefGoogle Scholar
Sly, A. (2011) Reconstruction for the Potts model. Ann. Probab. 39 1365–1406.CrossRefGoogle Scholar
Sly, A., Sun, N. and Zhang, Y. (2016) The number of solutions for random regular NAE-SAT. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 724–731.CrossRefGoogle Scholar
Zdeborová, L. and Krzakala, F. (2016) Statistical physics of inference: Thresholds and algorithms. Adv. Phys. 65 453–552.CrossRefGoogle Scholar