Explicit Bounds for Nondeterministically Testable Hypergraph Parameters

Abstract

In this note we give a new effective proof method for the equivalence of the notions of testability and nondeterministic testability for uniform hypergraph parameters. We provide the first effective upper bound on the sample complexity of any nondeterministically testable r-uniform hypergraph parameter as a function of the sample complexity of its witness parameter for arbitrary r. The dependence is of the form of an exponential tower function with the height linear in r. Our argument depends crucially on the new upper bounds for the r-cut norm of sampled r-uniform hypergraphs. We employ also our approach for some other restricted classes of hypergraph parameters, and present some applications.

A Characterization of Testable Hypergraph Properties

“…Karpinski and Markó [29] generalized the Lovász-Vesztergombi result to hypergraphs, also via the notion of (hyper-)graph limits. However, these proofs do not give an explicit bounds on the query complexity -this was achieved by Gishboliner and Shapira [32] for graphs and Karpinski and Markó [28] for hypergraphs.…”