April was a busy time in property testing with 9 (!) papers posted online this month. So let’s jump right in.

*Testing properties of graphs*

**Approximately Counting Triangles in Sublinear Time** by Talya Eden, Amit Levi, and Dana Ron (ECCC). Can we estimate the number of triangles in a sparse graph in sublinear time? When we can only query a vertex to learn its degree or its *i-*th neighbor, then the answer is no. But this paper shows that if we can also ask whether a pair of vertices are connected by an edge, then the answer is yes!

**Testing Cluster Structure of Graphs** by Artur Czumaj, Pan Peng, and Christian Sohler (arXiv). Another fundamental problem in the analysis of graphs is that of determining if the vertices of a graph can be partitioned into a small number of good clusters. This paper shows that the appropriate formalization of this problem can be solved in sublinear-time, and that in fact \(O(\sqrt{n})\) queries suffice to test whether a graph with \(n\) vertices can be partitioned into \(k = O(1)\) clusters.

*Testing properties of distributions*

**A Survey on Distribution Testing: Your Data is Big. But is it Blue?** by Clément Canonne (ECCC). As we have seen on this blog, there has been a lot of recent development in testing properties of distributions. This latest survey provides an up-to-date introduction to this research.

**Faster Algorithms for Testing under Conditional Sampling** by Moein Falahatgar, Ashkan Jafarpour, Alon Orlitsky, Venkatadheeraj Pichapathi, and Ananda Theertha Suresh (arXiv). One of the recent developments in distribution testing has been the discovery that natural conditional sampling models offer a surprising amount of algorithmic power. This paper gives yet more evidence of this fact, notably showing that testing the equivalence of two distributions with support size \(n\) can be done with roughly \(O(\log \log n)\) queries—a doubly-exponential improvement over the best possible algorithm for the same problem in the standard sampling model!

**Sampling Correctors** by Clément Canonne, Themis Gouleakis, and Ronitt Rubinfeld (arXiv). This paper initiates the study of a research direction related, but not exactly equivalent to distribution testing: given that we assume that the true distribution has some structural property (like monotonicity, for example), but that the samples we draw from this distribution may contain errors, can we somehow use these noisy samples to generate “corrected” samples from the original underlying distribution?

*Locally-testable codes*

**High rate locally-correctable and locally-testable codes with sub-polynomial query complexity** by Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf (ECCC). The central question in locally-testable error-correcting codes is to determine the tradeoff (if any!) that we must have between the number of queries require to test if an input is a codeword, the rate of the code, and the distance between the codewords. This paper shows that there are codes with constant rate and distance that can also be tested and decoded with sub-polynomial query complexity \(\exp(\tilde{O}(\sqrt{\log n}))\).

**Robust testing of lifted codes with applications to low-degree testing** by Alan Guo, Elad Haramaty, and Madhu Sudan (ECCC). Typically, we require a local test to reject inputs that are far from codewords with a reasonably large probability (that is: we want the testers to have good *soundness* properties). A stronger requirement might be to want the local view of a tester on those inputs to be *far* from the local view of any codeword with good probability. Testers that satisfy this property are called *robust*, and have many useful properties. This paper shows that natural testers of low-degree polynomials and their generalizations known as lifted codes are indeed robust testers.

*General property testing results*

**On Being Far from Far and on Dual Problems in Property Testing** by Roei Tell (ECCC). If I can efficiently test that an input has property \(\Pi\), can I also efficiently test if an input is far from having the property \(\Pi\)? At first glance, it seems that both these problems (that is, the direct and *dual* version of the problem of testing \(\Pi\)) are equivalent, but this is not always the case. This paper explores this subtle question and provides initial results for dual problems in testing properties of functions, graphs, and distributions.

**Trading query complexity for sample-based testing and multi-testing scalability** by Eldar Fischer, Oded Lachish, Yadu Vasudev (arXiv). What properties of combinatorial objects can we test in sublinear time when we have only sample access to it (instead of full query access)? This paper provides a general and powerful result showing that *every* property that can be tested non-adaptively with a constant number of queries can also be tested with a sublinear number of samples.