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 physicsbased 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 programmingbased 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 Gaussianweighted graph, and the number of large independent sets in a random dregular 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.
BibTeX  Entry
@InProceedings{hsieh_et_al:LIPIcs.CCC.2022.11,
author = {Hsieh, JunTing and Mohanty, Sidhanth and Xu, Jeff},
title = {{Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {11:111:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772419},
ISSN = {18688969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16573},
URN = {urn:nbn:de:0030drops165735},
doi = {10.4230/LIPIcs.CCC.2022.11},
annote = {Keywords: constraint satisfaction problems, certified counting, random graphs}
}
Keywords: 

constraint satisfaction problems, certified counting, random graphs 
Collection: 

37th Computational Complexity Conference (CCC 2022) 
Issue Date: 

2022 
Date of publication: 

11.07.2022 