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

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 nk/2 / ε2 constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+Ok(ε) fraction of constraints. Previously, this was known for even k, but for odd k one needed nk/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(nlceil k/2 rceil) and the third one cannot work when the number of constraints is o(nk/2 √log n). We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with nk/2 / ε2 constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.