Dall'Agnol, M. and Gur, T. and Lachish, Oded (2023) A structural theorem for local algorithms with applications to coding, testing, and privacy. SIAM Journal on Computing 52 (6), pp. 1413-1463. ISSN 0097-5397.
|
Text
adaptive-daisies.pdf - Author's Accepted Manuscript Available under License Creative Commons Attribution. Download (1MB) | Preview |
Abstract
We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 \log^2 q)}$ sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms. %Along the way, we obtain a sunflower-based combinatorial representation of robust local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. - We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an \emph{exponential} improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SICOMP 2021). - We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015), bypassing an exponential blowup caused by previous techniques in the case of adaptive testers. - We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal–Szemer\'{e}di theorem.
Metadata
| Item Type: | Article |
|---|---|
| Keyword(s) / Subject(s): | local decoding, algorithms, lower bounds |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Oded Lachish |
| Date Deposited: | 22 Feb 2024 10:46 |
| Last Modified: | 22 Feb 2024 14:01 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/53140 |
Statistics
Additional statistics are available via IRStats2.
Archive Staff Only (login required)
