This February, we had four exciting testing papers, covering domains from Boolean functions to distributions to graphs. Read on to catch up on the latest results!

**Property Testing of Joint Distributions using Conditional Samples**, by Rishiraj Bhattacharyya and Sourav Chakraborty (arXiv). This paper focuses on the problems of identity testing and independence testing for multivariate distributions in the conditional sampling model. Multivariate distribution testing has not previously been considered in the conditional sampling model. While previous algorithms for identity testing would generally carry over to this setting, the authors restrict themselves to algorithms where the queries are on subcubes of the domain. Interestingly, the authors show that the complexity of these problems is polynomial in the dimension (and that a polynomial dependence is necessary). That is, despite the “simple” structure of the queries, they are still powerful enough to avoid the curse of dimensionality which is present in vanilla multivariate distribution testing.

**An Improved Dictatorship Test with Perfect Completeness**, by Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari (arXiv, ECCC). A function is a dictator if it depends on exactly one variable. This paper provides an improved tester with perfect completeness for this property. In particular, they give a (non-adaptive) \(k\)-query test which never rejects a dictator function, and accepts a far-from-dictator function with probability at most \(\frac{2k +1}{2^k}\). This improves upon the previous best result, which accepts a far-from-dictator function with probability at most \(\frac{2k+3}{2^k}\).

**An Adaptivity Hierarchy Theorem for Property Testing**, by Clément L. Canonne and Tom Gur (arXiv, ECCC). This paper investigates the power of adaptivity in property testing. We most frequently think of algorithms as either adaptive or non-adaptive. In the former, an algorithm may specify each query after viewing the results of all previous queries, while in the latter, all the queries must be specified in advance. The authors focus on the natural interpolation, in which the algorithm is allowed to make queries in \(k\) rounds, where the queries in a round may depend on the result of queries in all previous rounds. They show that the power of an algorithm can shift dramatically with just one more round of adaptivity — there exist properties which can be tested with \(\tilde O(k)\) queries with \(k\) rounds of adaptivity, but require \(\Omega(n)\) queries with \(k-1\) rounds of adaptivity.

**Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness**, by Xi Chen, Erik Waingarten, and Jinyu Xie (arXiv). The main result in this paper is an improved lower bound for the well-studied problem of testing monotonicity of Boolean functions. The authors show that any two-sided and adaptive algorithm requires \(\tilde \Omega(n^{1/3})\) samples to test whether a function is monotone or far from monotone. This improves upon the previous result of \(\tilde \Omega(n^{1/4})\) by Belovs and Blais, which was the first to show that adaptive monotonicity testing requires polynomially-many queries. The construction of Belovs and Blais uses Talagrand’s random DNFs, for which they also showed \(\tilde O(n^{1/4})\) queries are sufficient. This paper analyzes an extension of this family, denoted as two-level Talagrand functions. The result leaves open the tantalizing question: does adaptivity help with monotonicity testing? Or does the complexity remain \(\tilde \Theta(\sqrt{n})\)? The authors also prove lower bounds for the problem of testing unateness, a problem of recent interest and a natural generalization of monotonicity.