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Linear Equations, Arithmetic Progressions and Hypergraph Property Testing

Summary: Linear Equations, Arithmetic Progressions and Hypergraph
Property Testing
Noga Alon
Asaf Shapira
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
D whose query complexity is polynomial in 1/ . For such k-graphs we say that PD (resp. P
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 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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics