2026-07-22T12:51:52Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18297 2024-03-06T11:01:32Z ddc:004 open_access

A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs d'Orsi, Tommaso Trevisan, Luca CSP k-XOR strong refutation sum-of-squares tensor graph hypergraph non-backtracking walk 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Tommaso d'Orsi and Luca Trevisan 2023 Is Part Of LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2023.27 urn:nbn:de:0030-drops-182979 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.27 eng https://creativecommons.org/licenses/by/4.0/legalcode