Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs

Yotam Dikstein, Irit Dinur, Alexander Lubotzky

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

Abstract

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [N] and let fs:s→Σ|sϵX be an ensemble of local functions, each defined over a subset s⊆ N. Suppose that we run the following so-called agreement test: choose a random pair of sets s1, s2 X that intersect on √k elements, and accept if fs1, fs2 agree on the elements in s1∩ s2. We denote the success probability of this test by Agree fs Given that Agree (fs)=ϵ > 0 is there a global function G:[N]→Σ such that fs=G|s| for a non-negligible fraction of s X We construct a family X of k-subsets of [N] such that |X|=O(N), and such that it satisfies the low acceptance agreement theorem. Namely, Agree (Formula Presented). A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The latter serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of sym-plectic HDXs.

Original languageEnglish
Title of host publicationProceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PublisherIEEE Computer Society
Pages826-861
Number of pages36
ISBN (Electronic)9798331516741
DOIs
StatePublished - 2024
Externally publishedYes
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

Bibliographical note

Publisher Copyright:
© 2024 IEEE.

Keywords

  • agreement
  • agreement testing
  • HDX
  • high dimensional expanders

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