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 language | English |
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Title of host publication | Proceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024 |
Publisher | IEEE Computer Society |
Pages | 826-861 |
Number of pages | 36 |
ISBN (Electronic) | 9798331516741 |
DOIs | |
State | Published - 2024 |
Externally published | Yes |
Event | 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024 - Chicago, United States Duration: 27 Oct 2024 → 30 Oct 2024 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |
Conference
Conference | 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024 |
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Country/Territory | United States |
City | Chicago |
Period | 27/10/24 → 30/10/24 |
Bibliographical note
Publisher Copyright:© 2024 IEEE.
Keywords
- agreement
- agreement testing
- HDX
- high dimensional expanders