A structural theorem for local algorithms with applications to coding, testing, and verification

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and probabilistically checkable proofs 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(q2 log 2 q)) sample complexity, following the definition of Goldreich and Ron [ACM Trans. Comput. Theory, 8 (2016), 7]. We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacypreserving 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 stateof-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish [SIAM J. Comput., 50 (2021), pp. 788-813]. 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 [Proceedings of the 56th Annual Symposium on Foundations of Computer Science, IEEE, 2015, pp. 1163-1182], 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 [Proceedings of the 8th Innovations in Theoretical Computer Science Conference, 2017, pp. 39:1-39:43; Comput. Complexity, 27 (2018), pp. 99-207] regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal-Szemeredi theorem.