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

TL;DR

This paper introduces a new non-backtracking matrix and an Ihara-Bass formula to improve polynomial-time strong refutation of random k-CSPs, especially for odd k, reducing the number of constraints needed.

Contribution

The paper develops a novel non-backtracking matrix and Ihara-Bass formula, enabling stronger refutation results for random k-CSPs with fewer constraints, overcoming previous limitations for odd k.

Findings

Achieved strong refutation with $n^{k/2} / \\epsilon^2$ constraints for odd k.

Improved the constraint threshold from $n^{k/2} (\\log n)^{O(1)}$ to $n^{k/2} / \\epsilon^2$ for odd k.

Extended techniques to semi-random k-CSPs with adversarial sign patterns.

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.