Linear equations, arithmetic progressions, and hypergraph property testing

Noga Alon, Asaf Shapira

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)177-216
Number of pages40
JournalTheory of Computing
Volume1
DOIs
StatePublished - Oct 12 2005
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computational Theory and Mathematics

Keywords

  • Additive number theory
  • Extremal problems
  • Hypergraphs
  • Linear algebra
  • Lower bounds
  • Property testing

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