Abstract
We define a novel notion of "nonbacktracking" matrix associated to any symmetric matrix, and we prove a "IharaBass" type formula for it.
We use this theory to prove new results on polynomialtime strong refutations of random constraint satisfaction problems with k variables per constraints (kCSPs). For a random kCSP 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 kCSP instance is quasirandom. Such certificate can come from a FeigeOfek 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 kCSP instances with n^{k/2} / ε² constraints in the semirandom 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 IharaBass Formula for NonBoolean Matrices and Strong Refutations of Random CSPs}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {27:127:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772822},
ISSN = {18688969},
year = {2023},
volume = {264},
editor = {TaShma, Amnon},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18297},
URN = {urn:nbn:de:0030drops182979},
doi = {10.4230/LIPIcs.CCC.2023.27},
annote = {Keywords: CSP, kXOR, strong refutation, sumofsquares, tensor, graph, hypergraph, nonbacktracking walk}
}
Keywords: 

CSP, kXOR, strong refutation, sumofsquares, tensor, graph, hypergraph, nonbacktracking walk 
Collection: 

38th Computational Complexity Conference (CCC 2023) 
Issue Date: 

2023 
Date of publication: 

10.07.2023 