The algorithmic aspects of the regularity lemma
Journal of Algorithms
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Extremal problems on set systems
Random Structures & Algorithms
Efficient Testing of Hypergraphs
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Testing subgraphs in directed graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Testing Subgraphs in Large Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Three theorems regarding testing graph properties
Random Structures & Algorithms
Regularity properties for triple systems
Random Structures & Algorithms
A characterization of easily testable induced subgraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Regularity lemma for k-uniform hypergraphs
Random Structures & Algorithms
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
An improved construction of progression-free sets
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property PD (resp. P*D) if it contains no copy (resp. induced copy) of D. Our goal in this paper is to classify the k-graphs D for which there are property-testers for testing PD and P*D whose query complexity is polynomial in 1/ε. For such k-graphs, we say that PD (or P*D) is easily testable.For P*D, we prove that aside from a single 3-graph, P*D is easily testable if and only if D is a single k-edge. For large k, we obtain stronger lower bounds than those obtained for the general case on the query complexity of testing P*D for any D other than the single k-edge. These bounds are proved by applying a more sophisticated technique than the basic one that works for all k. These results extend and improve previous results about graphs [5] and k-graphs [18].For PD, we show that for any k-partite k-graph D, PD, is easily testable, by giving an efficient one-sided error-property tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a property-tester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the k-graphs for which PD is easily testable, they are a natural extension of the previous results about graphs [1].Our proofs combine results and arguments from additive number theory, linear algebra and extremal hypergraph theory. We also develop new techniques, which are of independent interest. The first is a construction of a dense set of integers, which does not contain a subset that satisfies a certain set of linear equations. The second is an algebraic construction of certain extremal hypergraphs. We demonstrate the applicability of this last construction by resolving several cases of an open problem raised by Brown, Erdös and Sós in 1973. These two techniques have already been applied in two recent subsequent papers [6], [27].