I hope all of you are keeping safe and healthy in these difficult times. Thank heavens we have our research and our math to keep our sanity…

This month has seen two papers, one on testing variable partitions and one on distributed isomorphism testing.

**Learning and Testing Variable Partitions** by Andrej Bogdanov and Baoxiang Wang (arXiv). Consider a function \(f:\Sigma^n \to G\), where \(G\) is Abelian group. Let \(V\) denote the set of variables. The function \(f\) is \(k\)-separable if there is a partition \(V_1, V_2, \ldots, V_k\) of \(V\) such that \(f(V)\) can be expressed as the sum \(f_1(V_1) + f_2(V_2) + \ldots + f_k(V_k)\). This is an obviously natural property to study, though the specific application mentioned in the paper is high-dimensional reinforcement learning control. There are a number of learning results, but we’ll focus on the main testing result. The property of \(k\)-separability can be tested with \(O(kn^3/\varepsilon)\) queries, for \(\Sigma = \mathbb{Z}_q\) (and distance between functions is the usual Hamming distance). There is an analogous result (with different query complexity) for \(\Sigma = \mathbb{R}\). It is also shown that testing 2-separability requires \(\Omega(n)\) queries, even with 2-sided error. The paper, to its credit, also has empirical studies of the learning algorithm with applications to reinforcement learning.

**Distributed Testing of Graph Isomorphism in the CONGEST model **by Reut Levi and Moti Medina (arXiv). This result follows a recent line of research in distributed property testing algorithms. The main aim is to minimize the number of rounds of (synchronous) communication for a property testing problem. Let \(G_U\) denote the graph representing the distributive network. The aim is to test whether an input graph \(G_K\) is isomorphic to \(G_U\). The main property testing results are as follows. For the dense graph case, isomorphism can be property tested (with two-sided error) in \(O(D + (\varepsilon^{-1}\log n)^2) \) rounds, where \(D\) is the diameter of the graph and \(n\) is the number of nodes. (And, as a reader of this blog, you probably know what \(\varepsilon\) is already…). There is a standard \(\Omega(D)\) lower bound for distributed testing problems. For various classes of sparse graphs (like bounded-degree minor-free classes), constant time isomorphism (standard) property testers are known. This paper provides a simulation argument showing that standard/centralized \(q\)-query property testers can be implemented in the distributed model, in \(O(Dq)\) rounds (this holds for any property, not just isomorphism). Thus, these simulations imply \(O(D)\)-round property testers for isomorphism for bounded-degree minor-free classes.