A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
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We define a notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it. Previously, these notions were known only for symmetric 0/1 matrices. 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 / ϵ^2 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) / ϵ^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^⌈ k/2 ⌉) and the third one cannot work when the number of constraints is o(n^k/2√(log n)).
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