News for February 2015

The following three very interesting papers came out in the month of February.

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing by Jacob Fox, Janos Pach and Andrew Suk (arXiv). Testability of graph or hyper-graph properties is all about regularity. Alon and Shapira used regularity lemma for graphs to prove that every hereditary property of graphs is testable. Generalization of this result for hyper-graphs has been obtain recently. The upper bound on the query complexity for testing hereditary properties for graphs (or hyper-graphs) is a function that is independent of the size of the input but dependent on the promise parameter. The query complexity as a function of the promise parameter, ε, is the same as the function that appears in the regularity lemma (where ε is the approximation parameter), which is usually a tower of ε. In this paper it has been shown that for special hereditary graph properties (and for other generalized objects) the constant as a function of the approximation parameter is polynomial. Thus for these special classes of hereditary properties (for example H-freeness, where H is k-uniform) the query complexity for testing is a polynomial in the promise parameter.  This is an important step towards understanding the dependence of the query complexity for testing hereditary graph (and hyper-graph) properties on the promise parameter.

Proofs of Proximity for Context-Free Languages and Read-Once Branching Programs by Oded Goldreich, Tom Gur and Ron Rothblum (ECCC). In a recent paper of Gur and Rothblum, they studied the property testing model where the tester/verifier is given free access to a short proof in addition to the query access to the input. A related model called, interactive proof of proximity, by proposed by Rothblum, Vadhan and Wigderson where the verifier is allowed to interact with an all powerful prover.  There is a tradeoff between the size of the short proof or the amount of interaction and the amount of queries made to the input. In both these models the verifier is more powerful than in the traditional property testing setting  So for properties that are not testable in the standard property testing setting, understanding the query complexity in the stronger models is a natural question. In this paper context-free languages and small read-once branching programs has been studied.

Sample Complexity for Winner Prediction in Elections by Arnab Bhattacharyya and Palash Dey (arXiv). Number of random samples necessary to predict the winner in an election, where there are a number of contestants, is not only an interesting theoretical problem but also has lots of real life applications. Different voting rules may have different complexity in terms of number of samples required. In this paper a number of different standard voting rules have been considered and bounds for number of samples required has been presented. It is possibly the first time sampling has been analyzed in the context of social choice theory. This is indeed a area where a lot more theoretical works are expected to follow.

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