Happy near year, and best wishes to those close and \(\varepsilon\)-far! December concluded the year with 4 new preprints, spanning quite a lot of the property testing landscape:

**Testing Stability Properties in Graphical Hedonic Games**, by Hendrik Fichtenberger and Anja Rey (arXiv). The authors of this paper consider the problem of deciding whether a given *hedonic game* possesses some “coalition stability” in a property testing framework. Namely, recall that a hedonic game is a game where players (nodes) form coalitions (subsets of nodes) based on their individual preferences and local information about the considered coalition, thus resulting in a partition of the original graph.

Several notions exist to evaluate how good such a partition is, based on how “stable” the given coalitions are. This work focuses on hedonic games corresponding to bounded-degree graphs, introducing and studying the property testing question of deciding *(for several such notions of stability)* whether a given game admits a stable coalition structure, or is far from admitting such a partition.

**Spectral methods for testing cluster structure of graphs**, by Sandeep Silwal and Jonathan Tidor (arXiv). Staying among bounded-degree graphs, we turn to testing clusterability of graphs, the focus of this paper. Given an \(n\)-node graph \(G\) of degree at most \(d\) and parameters \(k, \phi\), say that \(G\) is \((k, \phi)\)-clusterable if it can be partitioned in \(k\) parts of inner conductance at least \(\phi\).

Analyzing properties of a random walk on \(G\), this work gives a bicriterion guarantee (\((k, \phi)\)-clusterable vs. \(\varepsilon\)-far from \((k, \phi^\ast)\)-clusterable, where \(\phi^\ast \approx \varepsilon^2\phi^2\)) for the case \(k=2\), improving on previous work by Czumaj, Peng, and Sohler’15.

We then switch from graphs to probability distributions with our third paper:

**Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction**, by Jayadev Acharya, Clément Canonne, and Himanshu Tyagi (arXiv). *(Disclaimer: I’m one of the authors.)* In this paper, the first of an announced series of three, the authors generalize the settings of two previous works we covered here and there to consider the general question of distribution testing and learning when the \(n\) i.i.d. samples are distributed among \(n\) players, which each can only communicate their sample to the central algorithm by respecting some pre-specified local information constraint *(e.g., privacy, or noise, or communication budget)*. This paper develops a general lower bound framework to study such questions, with a systematic focus on the power of public vs. private randomness between the \(n\) parties, and instantiate it to obtain tight bounds in the aforementioned locally private and communication-limited settings. (Spoiler: public randomness strictly helps, but not always.)

Finally, after games, graphs, and distributions, our fourth paper of the month concerns testing of functions:

**Partial Function Extension with Applications to Learning and Property Testing**, by Umang Bhaskar and Gunjan Kumar (arXiv). This work focuses on a problem quite related to property testing, that of partial function extension: given as input \(n\) pairs point/value from a purported function on a domain \(X\) of size \(|X| > n\), one is tasked with deciding whether there does exist (resp., with finding) a function \(f\) on \(X\) consistent with these \(n\) values which further satisfies a specific property, such as linearity or convexity. This is indeed very reminiscent of property testing, where one gets to query these \(n\) points and must decide (approximate) consistency with such a well-behaved function. Here, the authors study the computational hardness of this partial function extension problem, specifically for properties such as subadditivity and XOS (a sub-property of subadditivity); and as corollaries obtain new property testers for the classes of subadditive and XOS functions.

As usual, if you know of some work we missed from last December, let us know in the comments!