A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Verification

SIAM Journal on Computing, Volume 52, Issue 6, Page 1413-1463, December 2023.
Abstract. 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 [math] adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with [math] 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 privacy-preserving 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 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–Szemerédi theorem.