A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs

• URL: http://arxiv.org/abs/2204.10881v2
• Date: Sun, 14 May 2023 22:53:44 GMT
• Title: A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
• Authors: Tommaso d'Orsi, Luca Trevisan
• Abstract summary: We prove 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, we can efficiently compute a certificate that the optimum satisfies at most a $p+O_k(epsilon)$ fraction of constraints.
• Score: 4.023424709659175
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $n^{k/2} / \epsilon^2$ constraints, we can efficiently compute a certificate that the optimum satisfies at most a $p+O_k(\epsilon)$ fraction of constraints. Previously, this was known for even $k$, but for odd $k$ one needed $n^{k/2} (\log n)^{O(1)} / \epsilon^2$ 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 $o(n^{\lceil k/2 \rceil})$ and the third one cannot work when the number of constraints is $o(n^{k/2} \sqrt{\log n})$. We further apply our techniques to obtain a new PTAS finding assignments for $k$-CSP instances with $n^{k/2} / \epsilon^2$ constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

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