TY - JOUR
T1 - Linear equations, arithmetic progressions, and hypergraph property testing
AU - Alon, Noga
AU - Shapira, Asaf
N1 - Publisher Copyright:
© 2005 Noga Alon and Asaf Shapira.
PY - 2005/10/12
Y1 - 2005/10/12
N2 - For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property PD (or property P* D) if it contains no copy (or no 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 property PD (or property 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. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough k-graph. These results extend and improve the authors’ previous results about graphs (SODA 2004) and results by Kohayakawa, Nagle and Rödl on k-graphs (ICALP 2002). For PD, we show that for any k-partite k-graph D, property PD is easily testable. This is established by giving an efficient one-sided-error property-tester for PD, which improves the one obtained by Kohayakawa et al. 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 (Alon, 2002). Our proofs combine results and arguments from additive number theory, linear algebra, and extremal hypergraph theory. We also develop new techniques, which we believe 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. These techniques have already been applied in two papers under publication by the authors.
AB - For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property PD (or property P* D) if it contains no copy (or no 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 property PD (or property 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. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough k-graph. These results extend and improve the authors’ previous results about graphs (SODA 2004) and results by Kohayakawa, Nagle and Rödl on k-graphs (ICALP 2002). For PD, we show that for any k-partite k-graph D, property PD is easily testable. This is established by giving an efficient one-sided-error property-tester for PD, which improves the one obtained by Kohayakawa et al. 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 (Alon, 2002). Our proofs combine results and arguments from additive number theory, linear algebra, and extremal hypergraph theory. We also develop new techniques, which we believe 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. These techniques have already been applied in two papers under publication by the authors.
KW - Additive number theory
KW - Extremal problems
KW - Hypergraphs
KW - Linear algebra
KW - Lower bounds
KW - Property testing
UR - http://www.scopus.com/inward/record.url?scp=33847313522&partnerID=8YFLogxK
U2 - 10.4086/toc.2005.v001a009
DO - 10.4086/toc.2005.v001a009
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AN - SCOPUS:33847313522
SN - 1557-2862
VL - 1
SP - 177
EP - 216
JO - Theory of Computing
JF - Theory of Computing
ER -