News for November 2014

With the submission deadline for many conferences in November it is no surprise that we have good a number of nice articles related to property testing submitted in ArXiv and ECCC this month. Here they are:

Sunflowers and Testing Triangle-Freeness of Functions by Ishay Haviv and Ning Xie (arXiv). Testing triangle freeness is one of the central problems in property testing. The upper bound on the query complexity is obtained using the regularity lemma. The upper bound is constant that comes from the regularity lemma. The lower bound (in terms of the distance parameter, ε) is a polynomial in ε. In another line of recent works different variations of sunflower lemma has been studied.  Some important problems in complexity theory and algorithms, like matrix multiplication, has been connected to different variations of the sunflower lemma. In this work a new approach, using variations of sunflower lemma, is given to obtain lower bounds for triangle freeness. This work gives a new lower bound technique. It also reconfirms the need to understand the combinatorial problem of sunflower lemma and its variations.

Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts by Madhav Jha, C. Seshadhri and Ali Pinar (arXiv). Counting or approximately counting the number of occurrences of a particular subgraph in a given graph is a very important problem – not only as a mathematically interesting problem but also from the application point of view. The hard part is to design an algorithm that is provably good as well as empirically satisfactory. There is a huge line of research on this topic that solves this problem for a particular subgraph.  In this paper a 3-path sampling based algorithm is presented that can approximate the number of occurrences of any 4-vertex subgraph. Not only is the correctness and the running time of the algorithm is proved mathematically but the speed of the algorithm has been tested empirically also. This work should motivate us to design sub-linear algorithms that are both provably good as well as can be implementable.

Forrelation: A Problem that Optimally Separates Quantum from Classical Computing by Scott Aaronson and Andris Ambainis (arXiv). Understanding the power of quantum query over classical queries is well studied subject and has a long line of interesting results. A long standing open problem posed by Buhrman et al is related to the problem of what is the largest separation between classical query complexity and quantum query complexity.  In this paper it is shown that the Forrelation problem, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function, can be solved using 1 quantum query where as classically one needs Ω(√n/log n) number of queries. They also prove this is almost tight. This improves the previous know gap of 1 vs. Θ(n^{1/4}).

Query complexity in expectation by Jedrzej Kaniewski, Troy Lee and Ronald de Wolf (arXiv). In query complexity usually we consider the number of queries required for computing a particular given function f(x) exactly when x has to be queried. But if we are required to only output a value whose expected value (expectation taken over random coin tosses of the algorithm) is f(x), then what happens to the query complexity.  This natural question has been address in this paper for the first time. They prove that in this model both the randomized and quantum query complexity is equal to two natural definition of degree for the function f. This is very natural model of computing and seems like a very interesting direction to pursue. it should yield some nice results and applications in near future.

A Chasm Between Identity and Equivalence Testing with Conditional Queries by Jayadev Acharya, Clément L. Canonne and Gautam Kamath (arXiv). Recently a number of works have studied the power of conditional sampling. Conditional sampling in general gives lot more power compared to standard sampling model. For example testing identity of an unknown distribution, with support size n, can be done using constant number of conditional sampling, while in the standard sampling model Θ(√n) number of samples are required. For a related problem of testing equivalence of two unknown distributions, in the standard sampling model Θ(n^{2/3}) number of queries are required while only polylog(n) conditional samples are sufficient.  It was not known if this bound for conditional samples is tight for this problem of testing equivalence of unknown distribution. In this paper a lower bound of Ω(√(loglog n)) is proved.

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