Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 19, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Tommaso d'Orsi

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

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. This theory can be used to prove tight bounds on quadratic forms defined by random matrices --even when entries are not independent-- of -1/+1 vectors. We use these ideas to obtain new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraint (k-CSPs) that are tight up to constant factors.

Although, compared to previous works, 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 in our settings of interest and the third one cannot yield tight bounds in our settings of interest. Joint work with Luca Trevisan.

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