Agreement Theorems for High Dimensional Expanders in the Low Acceptance Regime: The Role of Covers

Yotam Dikstein, Irit Dinur

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

Abstract

Let X be a family of k-element subsets of [n] and let {fs:s→Σ: s∈ X} be an ensemble of local functions, each defined over a subset s⊂ [n]. Is there a global function G:[n]→Σ such that fs = G|s for all s∈ X ? An agreement test is a randomized property tester for this question. One such test is the V-test, that chooses a random pair of sets s1,s2∈ X with prescribed intersection size and accepts if fs1,fs2 agree on the elements in s1∩ s2. The low acceptance (or 1%) regime is concerned with the situation that the test succeeds with low but non-negligible probability Agree({fs}) ≥ ϵ>0. A "classical"low acceptance agreement theorem says Agree (Formula presented) Such statements are motivated by PCP questions. The case X= (k[n]) is well-studied and known as "direct product testing", which is related to the parallel repetition theorem. Finding sparser families X that satisfy (∗) is known as derandomized direct product testing. Prior to this work, the sparsest family satisfying (∗) had |X|≈ n25, and we show X with |X|≈ n2. We study the general behavior of high dimensional expanders with respect to agreement tests in the low acceptance regime. High dimensional expanders, even very sparse ones with |X|=O(n), are known to satisfy the high acceptance variant (where ϵ =1-o(1)). It has been an open challenge to analyze the low acceptance regime. Surprisingly, topological covers of X play an important role. We show that: If X has no connected covers, then (∗) holds, provided that X satisfies an additional expansion property, called swap cosystolic expansion. If X has a connected cover, then (∗) fails. If X has a connected cover (and swap-cosystolic-expansion), we replace (∗) by a statement that takes covers into account:(Formula presented). The property of swap-cosystolic-expansion holds for quotients of the Bruhat Tits buildings. As a corollary we derive (∗) for X being a spherical building, yielding a derandomized family with |X| ≈ n2. We also derive (∗∗) for LSV complexes X, for which |X|=O(n).

Original languageEnglish
Title of host publicationSTOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
EditorsBojan Mohar, Igor Shinkar, Ryan O�Donnell
Pages1967-1977
Number of pages11
ISBN (Electronic)9798400703836
DOIs
StatePublished - 10 Jun 2024
Event56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada
Duration: 24 Jun 202428 Jun 2024

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing

Conference

Conference56th Annual ACM Symposium on Theory of Computing, STOC 2024
Country/TerritoryCanada
CityVancouver
Period24/06/2428/06/24

Keywords

  • Agreement
  • Agreement Testing
  • Covers
  • Direct Product Testing
  • HDX
  • High Dimensional Expanders

All Science Journal Classification (ASJC) codes

  • Software

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