Helly-Type Theorems in Property Testing by Sourav Chakraborty, Rameshwar Pratap, Sasanka Roy, and Shubhangi Saraf (arXiv). Helly’s Theorem states that if every \(d+1\) sets in a family \(F\) of convex sets in \(\mathbb{R}^d\) have non-empty intersection, then the intersection of the whole family \(F\) is also non-empty. This paper establishes a new connection between Helly’s theorem and clustering problems. Specifically, extensions of Helly’s theorem are used to analyze algorithms that test whether a set of points can be partitioned into a small number of clusters or not.

On active and passive testing by Noga Alon, Rani Hod, and Amit Weinstein (arXiv). In the standard property testing setting, the tester is able to query any input it chooses. This model is unrealistic in some settings, where instead it is more appropriate to consider models where the tester can only choose queries from a restricted set of the inputs (active testing) or where the tester only sees bits sampled at random from the input (passive testing). This paper gives new upper and lower bounds on the number of queries required to test many natural properties of boolean functions—including juntas, partially symmetric functions, and low degree polynomials—in these two settings.

Explicit Maximally Recoverable Codes with Locality by Parikshit Gopalan, Cheng Huang, Bob Jenkins, and Sergey Yekhanin (ECCC). The central goal in the study of locally decodable codes is to maintain the rate and error tolerance while adding the condition that we can recover any bit of the original input with a small number of queries to the codeword. This paper considers and makes progress on an interesting relaxation of this problem: here the goal is to maintain local decoding when a small number of errors are introduced, while allowing more queries to be made to decode input bits when more errors occur.

In June, we had a property testing workshop at Haifa where a lot of recent PT work was discussed. Check out the post on that. June also saw some interesting developments on testing affine-invariant properties, on testing sub-properties, on local computation algorithms, and on PCP constructions.

TESTING AFFINE-INVARIANT PROPERTIES

Over the last few years, there has been much progress in determining which affine-invariant properties of boolean functions can be tested with a constant number of queries. In Estimating the distance from testable affine-invariant properties (arXiv, ECCC), Hamed Hatami and Shachar Lovett show that for every such property, we can not only test it but also estimate the distance to the property with a constant number of queries.

The function linear isomorphism testing problem asks: How many queries do we need to test if a given (unknown) function is equivalent, up to a linear transformation of the input space, to some (known) function \(f\)? The answer to this question depends on the choice of the function \(f\). Elena Grigorescu, Karl Wimmer, and Ning Xie, in Tight lower bounds for testing linear isomorphism (ECCC) and Abhishek Bhrushundi, in the concurrent and independent paper On testing bent functions (ECCC), show that the query complexity for testing linear isomorphism is maximized when \(f\) is the Inner Product function. Interestingly, the proofs of this result (and other more general results) are obtained using completely different methods: Elena, Karl, and Ning prove their lower bounds using the communication complexity method, while Abhishek’s proof is obtained by studying the parity decision tree complexity of boolean functions.

TESTING SUB-PROPERTIES

One counter-intuitive aspect of property testing is that the query complexity for testing \(P\) does not in general imply anything about the query complexity for testing a sub-property \(P’ \subseteq P\). For example, while we can test halfspaces (aka, linear threshold functions) with a constant number of queries, testing the subclass of signed majorities is known to require \(\Omega(\log n)\) queries, and in fact the best-known algorithm for this task is a non-adaptive tester that requires \(O(\sqrt{n})\) queries. In Exponentially improved algorithms and lower bounds for testing signed majorities (ECCC), Dana Ron and Rocco Servedio dramatically improve both the upper and lower bounds: they show that non-adaptive algorithms for testing signed majorities require \(\mathrm{poly}(n)\) queries and that signed majorities can be tested by an adaptive algorithm that requires only \(\mathrm{polylog}(n)\) queries.

Another phenomenon that appears in some natural properties is that while a property \(P\) requires many queries to test, it can be partitioned into (slightly) smaller properties \(P’\) that can each be tested with a constant number of queries. It is natural to ask whether this is a universal phenomenon. In Some properties are not even partially testable (arXiv, ECCC), Eldar Fischer, Yonatan Goldhirsh, and Oded Lachish show that it is not: they show that there are properties \(P\) for which every (large enough) sub-property of \(P\) requires a large number of queries to test.

LOCAL COMPUTATION AND PCP CONSTRUCTIONS

A notion that is very closely related to property testing is that of local computation
algorithms: algorithms that, as in the property testing setting, aim to compute the solution of a problem in sublinear time by querying as few bits of the input as possible. In A Local Computation Approximation Scheme to Maximum Matching (arXiv), Yishay Mansour and Shai Vardi give a new local computation algorithm for obtaining a \((1-\epsilon)\)-approximation to the maximum matching in bounded-degree graphs.

The notion of Probabilistically Checkable Proofs (PCPs) is also closely related to property testing, where now the input to the tester is a string \(x\) and a purported proof that \(x\) satisfies some property \(P\); the tester must verify the correctness of the proof while examining as few bits of \(x\) and of the proof as possible. A long-standing open problem in this area is to understand the best possible trade-offs between the query complexity and the length of proofs for PCP constructions. In Constant rate PCPs for circuit-SAT with sublinear query complexity (ECCC), Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, and Henning Stichtenoth give a verifier for a special case of PCPs that obtains a sublinear query complexity with proofs that have length only linear in the size of the input.