This month saw four papers. One on group testing, another on distribution testing, yet another which makes progress on testing problems on decision trees and the last one on graph property testing. Without further ado, let’s dive in.

**Group Testing with Non-identical Infection Probabilities** by Mustafa Doger, Sennur Ulukus (arXiv) Consider the classic group testing problem. Here the setup is the following. You are given a bunch of individuals from a population \(\mathcal{P}\). You have an infection vector which records the infection status of each individual in the population where the \(i\)-th individual is infected with probability \(p_i\). You want to recover all the infected individuals. You are allowed to group individuals together and you can test the entire group in a single shot. If the group tests negative, you are happy all the tested individuals are off the hook. Otherwise, if the group tests positive, you need more tests for further classification. This paper proposes a greedy way to build pools of individuals you would test. The pools are built adaptively: as in future pools are built using the knowledge of how the preceding tests fared. The key result in the paper upperbounds the number of tests performed in terms of the entropy of the infection vector.

**Uniformity Testing in the Shuffle Model: Simpler, Better, Faster** by Clément L. Canonne, Hongyi Lyu (arXiv) Differentially private distribution testing as a research area has been gathering momentum steadily over the last few years. If you read our last month’s post, you might recall there are a wide variety of models of DP each corresponding to a different “threat model”. The most stringent among the most explored models is the “local model”, the least stringent being the “central model” and there is an intermediate threat model, the so called “shuffle model“. This paper simplifies the analysis of uniformity testing algorithm under the shuffle model and presents an algorithm with sample complexity \(O(k^{3/4})\) for testing uniformity over a support of size \([k]\).

**On Learning and Testing Decision Tree** by Nader H. Bshouty, Catherine A. Haddad-Zaknoon (arXiv) In our December 2020 post, we covered a result of Blanc et al., which proves the following: Suppose you are given a boolean function \(f\) and the property \(\mathcal{P}\) of size-\(s\) decision trees. The result of Blanc et al gives you a function \(g \in \mathcal{P}\) with \(dist(f,g) = O(dist (f, \mathcal{P}))\) where \(g \in \) is guaranteed to have decision tree complexity \(s^{O(\log^2 s)}\). This result implies a bi-criteria tester for the following property: is \(f \in \mathcal{P}\) or is \(f\) \(\varepsilon\)-far from having decision tree complexity \(\phi(s) = s^{O(\log^3 s)}\). The current paper improves this result by presenting a property tester with \(\phi(s) = s^{O(\log^2 s)}\).

**The complexity of testing all properties of planar graphs, and the role of isomorphism** by Sabyasachi Basu, Akash Kumar, C. Seshadhri (arXiv) *(Disclaimer: I am one of the authors of this paper)*. This paper presents a result that I, in my biased opinion, find interesting. So, here is the setup. You are given a bounded degree planar graph. And I cook up some God-forsaken property and ask you to test it. Turns out, no matter how devilishly I cooked up the property, you can test in with \(\exp(O(\varepsilon^{-2}))\) queries. The nice happenstance is that you also have a matching lower bound of \(\exp(\Omega(\varepsilon^{-2}))\) queries! And interestingly, this lower bound is witnessed by the very natural property of testing isomorphism to a fixed graph which means that isomorphism is the hardest property of planar graphs.