Document Open Access Logo

Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance

Authors Jun-Ting Hsieh, Sidhanth Mohanty, Jeff Xu

PDF
Thumbnail PDF

File

LIPIcs.CCC.2022.11.pdf
  • Filesize: 0.82 MB
  • 18 pages

Document Identifiers

Author Details

Jun-Ting Hsieh
  • Carnegie Mellon University, Pittsburgh, PA, USA
Sidhanth Mohanty
  • University of California at Berkeley, CA, USA
Jeff Xu
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Hsieh, Jun-Ting: Supported by NSF CAREER Award #2047933.
  • Mohanty, Sidhanth: Supported by Google PhD Fellowship.
  • Xu, Jeff: Supported by NSF CAREER Award #2047933.

Acknowledgements

We would like to thank Pravesh Kothari, Prasad Raghavendra, and Tselil Schramm for their encouragement and thorough feedback on an earlier draft. We would also like to thank Tselil Schramm for enlightening discussions on refuting random CSPs.

Cite AsGet BibTex

Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu. Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.11

Abstract

An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: What does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1) Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random d-regular graph. 2) Clusters. For Boolean 3CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3) Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of +1s deviates significantly from 50%. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • constraint satisfaction problems
  • certified counting
  • random graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Dimitris Achlioptas and Amin Coja-Oghlan. Algorithmic barriers from phase transitions. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 793-802. IEEE, 2008. Google Scholar
  2. Dimitris Achlioptas and Cristopher Moore. The asymptotic order of the random k-SAT threshold. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 779-788. IEEE, 2002. Google Scholar
  3. Dimitris Achlioptas and Yuval Peres. The threshold for random k-sat is 2^k (ln 2 - o(k)). In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 223-231, 2003. Google Scholar
  4. Dimitris Achlioptas and Federico Ricci-Tersenghi. On the solution-space geometry of random constraint satisfaction problems. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 130-139, 2006. Google Scholar
  5. Sarah R Allen, Ryan O'Donnell, and David Witmer. How to refute a random CSP. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 689-708. IEEE, 2015. Google Scholar
  6. Noga Alon. Independent sets in regular graphs and sum-free subsets of finite groups. Israel journal of mathematics, 73(2):247-256, 1991. Google Scholar
  7. Noga Alon. Perturbed identity matrices have high rank: Proof and applications. Combinatorics, Probability & Computing, 18(1-2):3, 2009. Google Scholar
  8. Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 171-180, 2010. Google Scholar
  9. Boaz Barak, Siu On Chan, and Pravesh K Kothari. Sum of squares lower bounds from pairwise independence. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 97-106, 2015. Google Scholar
  10. Alexander Barvinok. Combinatorics and complexity of partition functions, volume 9. Springer, 2016. Google Scholar
  11. Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Stefankovic. Approximation via correlation decay when strong spatial mixing fails. SIAM Journal on Computing, 48(2):279-349, 2019. Google Scholar
  12. Béla Bollobás. The independence ratio of regular graphs. Proceedings of the American Mathematical Society, pages 433-436, 1981. Google Scholar
  13. Andries E Brouwer and Willem H Haemers. Spectra of graphs. Springer Science & Business Media, 2011. Google Scholar
  14. Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. Classifying the complexity of constraints using finite algebras. SIAM journal on computing, 34(3):720-742, 2005. Google Scholar
  15. Amin Coja-Oghlan. Belief propagation guided decimation fails on random formulas. Journal of the ACM (JACM), 63(6):1-55, 2017. Google Scholar
  16. Amin Coja-Oghlan, Andreas Goerdt, and André Lanka. Strong refutation heuristics for random k-SAT. Combinatorics, Probability & Computing, 16(1):5, 2007. Google Scholar
  17. Amin Coja-Oghlan, Noëla Müller, and Jean B Ravelomanana. Belief Propagation on the random k-SAT model. arXiv preprint, 2020. URL: http://arxiv.org/abs/2011.02303.
  18. Amin Coja-Oghlan and Konstantinos Panagiotou. Going after the k-SAT threshold. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 705-714, 2013. Google Scholar
  19. Amin Coja-Oghlan and Konstantinos Panagiotou. The asymptotic k-SAT threshold. Advances in Mathematics, 288:985-1068, 2016. Google Scholar
  20. Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. Statistical physics approaches to Unique Games. arXiv preprint arXiv:1911.01504, 2019. Google Scholar
  21. Amit Daniely, Nati Linial, and Shai Shalev-Shwartz. From average case complexity to improper learning complexity. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 441-448, 2014. Google Scholar
  22. Jian Ding, Allan Sly, and Nike Sun. Proof of the satisfiability conjecture for large k. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 59-68, 2015. Google Scholar
  23. Martin Dyer and Catherine Greenhill. On Markov chains for independent sets. Journal of Algorithms, 35(1):17-49, 2000. Google Scholar
  24. Paul Erdős and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. In Colloquia Mathematica Societatis Janos Bolyai 10. Infinite and Finite Sets, Keszthely (Hungary). Citeseer, 1973. Google Scholar
  25. Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 534-543, 2002. Google Scholar
  26. Uriel Feige and Eran Ofek. Spectral techniques applied to sparse random graphs. Random Structures & Algorithms, 27(2):251-275, 2005. Google Scholar
  27. Uriel Feige and Eran Ofek. Easily refutable subformulas of large random 3CNF formulas. Theory of Computing, 3(1):25-43, 2007. Google Scholar
  28. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Fast sampling and counting k-SAT solutions in the local lemma regime. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 854-867, 2020. Google Scholar
  29. Weiming Feng, Kun He, and Yitong Yin. Sampling Constraint Satisfaction Solutions in the Local Lemma Regime. arXiv preprint, 2020. URL: http://arxiv.org/abs/2011.03915.
  30. Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Kuan Yang. Counting solutions to random CNF formulas. arXiv preprint, 2019. URL: http://arxiv.org/abs/1911.07020.
  31. Mrinalkanti Ghosh, Fernando Granha Jeronimo, Chris Jones, Aaron Potechin, and Goutham Rajendran. Sum-of-squares lower bounds for Sherrington-Kirkpatrick via planted affine planes. arXiv preprint, 2020. URL: http://arxiv.org/abs/2009.01874.
  32. Andreas Goerdt and André Lanka. Recognizing more random unsatisfiable 3-SAT instances efficiently. Electronic Notes in Discrete Mathematics, 16:21-46, 2003. Google Scholar
  33. Oded Goldreich. Candidate one-way functions based on expander graphs. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 76-87. Springer, 2011. Google Scholar
  34. Dima Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. computational complexity, 10(2):139-154, 2001. Google Scholar
  35. Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4):798-859, 2001. Google Scholar
  36. Samuel B Hopkins and David Steurer. Efficient Bayesian estimation from few samples: community detection and related problems. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 379-390. IEEE, 2017. Google Scholar
  37. Vishesh Jain, Frederic Koehler, and Andrej Risteski. Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1226-1236, 2019. Google Scholar
  38. Vishesh Jain, Huy Tuan Pham, and Thuy Duong Vuong. Towards the sampling Lovász Local Lemma. arXiv preprint, 2020. URL: http://arxiv.org/abs/2011.12196.
  39. Vishesh Jain, Huy Tuan Pham, and Thuy-Duong Vuong. On the sampling Lovász Local Lemma for atomic constraint satisfaction problems. arXiv preprint, 2021. URL: http://arxiv.org/abs/2102.08342.
  40. Jeff Kahn. An entropy approach to the hard-core model on bipartite graphs. Combinatorics, Probability & Computing, 10(3):219, 2001. Google Scholar
  41. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775, 2002. Google Scholar
  42. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007. Google Scholar
  43. Pravesh Kothari, Ryan O'Donnell, and Tselil Schramm. SOS lower bounds with hard constraints: think global, act local. arXiv preprint, 2018. URL: http://arxiv.org/abs/1809.01207.
  44. Pravesh K Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 132-145, 2017. Google Scholar
  45. Florent Krzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318-10323, 2007. Google Scholar
  46. Dmitriy Kunisky and Afonso S Bandeira. A tight degree 4 sum-of-squares lower bound for the Sherrington-Kirkpatrick Hamiltonian. Mathematical Programming, pages 1-39, 2020. Google Scholar
  47. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. A deterministic algorithm for counting colorings with 2Δ colors. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1380-1404. IEEE, 2019. Google Scholar
  48. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. The Ising partition function: Zeros and deterministic approximation. Journal of Statistical Physics, 174(2):287-315, 2019. Google Scholar
  49. Robert McEliece, Eugene Rodemich, Howard Rumsey, and Lloyd Welch. New upper bounds on the rate of a code via the delsarte-macwilliams inequalities. IEEE Transactions on Information Theory, 23(2):157-166, 1977. Google Scholar
  50. Stephan Mertens, Marc Mézard, and Riccardo Zecchina. Threshold values of random k-sat from the cavity method. Random Structures & Algorithms, 28(3):340-373, 2006. Google Scholar
  51. Marc Mézard, Thierry Mora, and Riccardo Zecchina. Clustering of solutions in the random satisfiability problem. Physical Review Letters, 94(19):197205, 2005. Google Scholar
  52. Marc Mézard, Giorgio Parisi, and Riccardo Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297(5582):812-815, 2002. Google Scholar
  53. Marc Mézard and Riccardo Zecchina. Random k-satisfiability problem: From an analytic solution to an efficient algorithm. Physical Review E, 66(5):056126, 2002. Google Scholar
  54. Sidhanth Mohanty, Prasad Raghavendra, and Jeff Xu. Lifting sum-of-squares lower bounds: degree-2 to degree-4. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 840-853, 2020. Google Scholar
  55. Ankur Moitra. Approximate counting, the Lovász local lemma, and inference in graphical models. Journal of the ACM (JACM), 66(2):1-25, 2019. Google Scholar
  56. Andrea Montanari. Optimization of the sherrington-kirkpatrick hamiltonian. In Proceedings of the 60th IEEE Symposium on Foundations of Computer Science, pages 1417-1433, 2019. Google Scholar
  57. Andrea Montanari, Federico Ricci-Tersenghi, and Guilhem Semerjian. Clusters of solutions and replica symmetry breaking in random k-satisfiability. Journal of Statistical Mechanics: Theory and Experiment, 2008(04):P04004, 2008. Google Scholar
  58. Andrea Montanari and Subhabrata Sen. Semidefinite programs on sparse random graphs and their application to community detection. In Proceedings of the 48th annual ACM Symposium on Theory of Computing, pages 814-827, 2016. Google Scholar
  59. Andrea Montanari and Devavrat Shah. Counting good truth assignments of random k-SAT formulae. arXiv preprint, 2006. URL: http://arxiv.org/abs/cs/0607073.
  60. Dmitry Panchenko. Spin glass models from the point of view of spin distributions. Annals of Probability, 41(3A):1315-1361, 2013. Google Scholar
  61. Giorgio Parisi. Infinite number of order parameters for spin-glasses. Physical Review Letters, 43(23):1754, 1979. Google Scholar
  62. Giorgio Parisi. A sequence of approximated solutions to the SK model for spin glasses. Journal of Physics A: Mathematical and General, 13(4):L115, 1980. Google Scholar
  63. Viresh Patel and Guus Regts. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing, 46(6):1893-1919, 2017. Google Scholar
  64. Han Peters and Guus Regts. On a conjecture of Sokal concerning roots of the independence polynomial. The Michigan Mathematical Journal, 68(1):33-55, 2019. Google Scholar
  65. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 245-254, 2008. Google Scholar
  66. Prasad Raghavendra, Satish Rao, and Tselil Schramm. Strongly refuting random CSPs below the spectral threshold. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 121-131, 2017. Google Scholar
  67. Andrej Risteski. How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods. In Conference on Learning Theory, pages 1402-1416. PMLR, 2016. Google Scholar
  68. Andrej Risteski and Yuanzhi Li. Approximate maximum entropy principles via goemans-williamson with applications to provable variational methods. Advances in Neural Information Processing Systems, 29:4628-4636, 2016. Google Scholar
  69. Thomas J Schaefer. The complexity of satisfiability problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 216-226, 1978. Google Scholar
  70. Grant Schoenebeck. Linear level Lasserre lower bounds for certain k-CSPs. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 593-602. IEEE, 2008. Google Scholar
  71. David Sherrington and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters, 35(26):1792, 1975. Google Scholar
  72. Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 361-369. IEEE, 2012. Google Scholar
  73. Michel Talagrand. The Parisi formula. Annals of mathematics, pages 221-263, 2006. Google Scholar
  74. Eric Vigoda. A note on the Glauber dynamics for sampling independent sets. the electronic journal of combinatorics, 8(1):R8, 2001. Google Scholar
  75. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 140-149, 2006. Google Scholar
  76. Nicholas C Wormald. Models of random regular graphs. London Mathematical Society Lecture Note Series, pages 239-298, 1999. Google Scholar
  77. Yufei Zhao. The number of independent sets in a regular graph. Combinatorics, Probability and Computing, 19(2):315-320, 2010. Google Scholar
  78. Dmitriy Zhuk. A Proof of the CSP Dichotomy Conjecture. Journal of the ACM (JACM), 67(5):1-78, 2020. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact