A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Verification - Dall'Agnol, Marcel; Gur, Tom; Lachish, Oded (arXiv.org, 2023)
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. 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 (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) by extending their main result to 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-Szemerédi theorem.