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Volume 1 (2005) Article 9 pp. 177-216
Linear Equations, Arithmetic Progressions and Hypergraph Property Testing
Received: January 12, 2005
Published: October 12, 2005
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Keywords: Property Testing, Hypergraphs, Lower Bounds, Additive Number Theory, Linear Algebra, Extremal Problems
ACM Classification: G.2.2, F.2.2
AMS Classification: 05C65, 68R10

Abstract: [Plain Text Version]

For a fixed $k$-uniform hypergraph $D$ ($k$-graph for short, $k \geq 3$), we say that a $k$-graph $H$ satisfies property ${\cal P}_D$ (or property ${\cal P}^*_D$) 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 ${\cal P}_D$ and ${\cal P}^*_D$ whose query complexity is polynomial in $1/\epsilon$. For such $k$-graphs we say that property ${\cal P}_D$ (or property ${\cal P}^*_D$) is easily testable.

For ${\cal P}^*_D$, we prove that aside from a single $3$-graph, ${\cal P}^*_D$ 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 ${\cal P}_D$, we show that for any $k$-partite $k$-graph $D$, property ${\cal P}_D$ is easily testable. This is established by giving an efficient one-sided-error property-tester for ${\cal P}_D$, 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 ${\cal P}_D$ is not easily testable. Though our results do not supply a complete characterization of the $k$-graphs for which ${\cal P}_D$ 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.