News for March 2018

March has been a relatively slow month for property testing, with 3 works appearing online. (If you notice we missed some, please leave a comment pointing it out)

Edge correlations in random regular hypergraphs and applications to subgraph testing, by Alberto Espuny Díaz, Felix Joos, Daniela Kühn, and Deryk Osthus (arXiv). While testing subgraph-freness in the dense graph model is now well-understood, after a series of works culminating in a complete characterization of the testing problems which admit constant-query testers, the corresponding question for hypergraphs is far from resolved. In this paper, the authors develop new techniques for the study of study of random regular hypergraphs, which imply new testing results for subhypergraph-freeness testing, improving on the state-of-the-art for some parameter regimes (e.g., when the input graph satisfies some average-degree condition).

Back from hypergraphs to graphs, we also have:
The Subgraph Testing Model, by Oded Goldreich and Dana Ron (ECCC). Here, the authors introduce a new model for property testing of graphs, where the goal is to test if an unknown subgraph \(F\) of an explicitly given graph \(G=(V,E)\) satisfies the desired property. The testing algorithm is provided access to \(F\) via membership queries, i.e., through query access to the indicator function \(\mathbf{1}_F\colon E \to \{0,1\}\). (In some very hazy sense, this is reminiscent of the active learning or testing frameworks, where one gets more or less free access to unlabeled data but pays to see their label.) As a sample of the results obtained, the paper establishes that this new model and the bounded-degree graph model are incomparable: there exist properties easier to test in one model than the other, and vice-versa — and some properties equally easy to test in both.

And finally, to drive home the point that “models matter a lot,” we have our third paper:
Every set in P is strongly testable under a suitable encoding, by Irit Dinur, Oded Goldreich, and Tom Gur (ECCC). It is known that the choice of representation of the objects has a large impact in property testing: for instance, the complexity of testing a given property can change drastically between the dense and bounded-degree graph models. This work provides another example of such a strong dependence on the representation: while membership to some sets in \(P\) is known to be hard to test, the authors here prove that, for every set \(S\in P\), there exists a (polynomial-time, invertible) encoding \(E_S\colon \{0,1\}^\ast\to \{0,1\}^\ast\) such that testing membership to \(S\) under this encoding is easy. (They actually show even stronger a statement: namely, that under this encoding the set admits a “proximity-oblivious tester,” that is a constant-query testing algorithm which rejects with probability function of the distance to \(S\).)

Finally, on a non-property testing note: Edith Cohen, Vitaly Feldman, Omer Reingold, and Ronitt Rubinfeld recently wrote a pledge for inclusiveness in the TCS community, available here: https://www.gopetition.com/petitions/a-pledge-for-inclusiveness-in-toc.html
If you haven’t seen it already, we encourage you to read it.

Update: Fixed a mistake in the overview of the second paper; as pointed out by Oded in the comments, the main comparison was between the new model and the bounded-degree graph model, not the dense graph one.

2 thoughts on “News for March 2018

  1. Oded Goldreich

    There is a typo in Clement’s review. Although the paper “The subgraph testing model” does contain a remark about the relation to the dense model (which may be viewed as the case that the base graph is the complete graph), the focus is on the relation of the new model to the bounded-degree graph model.

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