Tight Bounds for the Zig-Zag Product

Gil Cohen, Itay Cohen, Gal Maor

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The Zig-Zag product of two graphs, Z= GzH, was introduced in the seminal work of Reingold, Vadhan, and Wigderson (Ann. of Math. 2002) and has since become a pivotal tool in theoretical computer science. The classical bound, which is used throughout, states that the spectral expansion of the Zig-Zag product can be bounded roughly by the sum of the spectral expansions of the individual graphs, Ω z≤ΩHG. In this work we derive, for every (vertex-transitive) c-regular graph H on d vertices, a tight bound for Ω z by taking into account the entire spectrum of H. Our work reveals that the bound, which holds for every graph G, is precisely the minimum value of the function (Equation Presented) in the domain (c2, ∞), where H(x) is the characteristic polynomial of H2. As a consequence, we establish that Zig-Zag products are indeed intrinsically quadratic away from being Ramanujan. We further prove tight bounds for the spectral ex-pansion of the more fundamental replacement product. Our lower bounds are based on results from analytic combinatorics, and we make use of finite free probability to prove their tightness. In a broader context, our work uncovers intriguing links between the two fields and these well-studied graph operators.

Original languageEnglish
Title of host publicationProceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PublisherIEEE Computer Society
Pages1470-1499
Number of pages30
ISBN (Electronic)9798331516741
DOIs
StatePublished - 2024
Event65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024 - Chicago, United States
Duration: 27 Oct 202430 Oct 2024

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024
Country/TerritoryUnited States
CityChicago
Period27/10/2430/10/24

Funding

FundersFunder number
European Research Council949499
Israel Science Foundation1569/18

    Keywords

    • combinatorics
    • pseudorandomness and derandomization
    • spectral methods

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