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 ... more >>>

We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string ... more >>>

The determinant is a canonical VBP-complete polynomial in the algebraic complexity setting. In this work, we introduce two variants of the determinant polynomial which we call $StackDet_n(X)$ and $CountDet_n(X)$ and show that they are VP and VNP complete respectively under $p$-projections. The definitions of the polynomials are inspired by a ... more >>>