License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2023.27
URN: urn:nbn:de:0030-drops-182979
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18297/
d'Orsi, Tommaso ;
Trevisan, Luca
A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
Abstract
We define a novel notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it.
We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and n^{k/2} / ε² constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+O_k(ε) fraction of constraints.
Previously, this was known for even k, but for odd k one needed n^{k/2} (log n)^{O(1)} / ε² random constraints to achieve the same conclusion.
Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck’s inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n^⌈k/2⌉) and the third one cannot work when the number of constraints is o(n^{k/2} √{log n}).
We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with n^{k/2} / ε² constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.
BibTeX - Entry
@InProceedings{dorsi_et_al:LIPIcs.CCC.2023.27,
author = {d'Orsi, Tommaso and Trevisan, Luca},
title = {{A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18297},
URN = {urn:nbn:de:0030-drops-182979},
doi = {10.4230/LIPIcs.CCC.2023.27},
annote = {Keywords: CSP, k-XOR, strong refutation, sum-of-squares, tensor, graph, hypergraph, non-backtracking walk}
}
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Keywords: |
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CSP, k-XOR, strong refutation, sum-of-squares, tensor, graph, hypergraph, non-backtracking walk |
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Collection: |
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38th Computational Complexity Conference (CCC 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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10.07.2023 |
