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

Authors Jun-Ting Hsieh, Sidhanth Mohanty, Jeff Xu

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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 As Get 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

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