Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

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

Mittagsseminar Talk Information

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

Duration: 30 minutes

Location: CAB G51

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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