Trading query complexity for sample-based testing and multi-testing scalability

Abstract: We show here that every non-adaptive property testing algorithm making a constant number of queries, over a fixed alphabet, can be converted to a sample-based (as per [Goldreich and Ron, 2015]) testing algorithm whose average number of queries is a fixed, smaller than $1$, power of $n$. Since the query distribution of the sample-based algorithm is not dependent at all on the property, or the original algorithm, this has many implications in scenarios where there are many properties that need to be tested for concurrently, such as testing (relatively large) unions of properties, or converting a Merlin-Arthur Proximity proof (as per [Gur and Rothblum, 2013]) to a proper testing algorithm. The proof method involves preparing the original testing algorithm for a combinatorial analysis, which in turn involves a new result about the existence of combinatorial structures (essentially generalized sunflowers) that allow the sample-based tester to replace the original constant query complexity tester.