eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-07-10
27:1
27:16
10.4230/LIPIcs.CCC.2023.27
article
A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
d'Orsi, Tommaso
1
Trevisan, Luca
2
Department of Computer Science, ETH Zürich, Switzerland
Department of Computing Sciences, Bocconi University, Milano, Italy
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol264-ccc2023/LIPIcs.CCC.2023.27/LIPIcs.CCC.2023.27.pdf
CSP
k-XOR
strong refutation
sum-of-squares
tensor
graph
hypergraph
non-backtracking walk