Explicit Bounds for Nondeterministically Testable Hypergraph Parameters

TL;DR

This paper establishes effective bounds on the sample complexity of nondeterministically testable hypergraph parameters, linking it to the complexity of witness parameters with a tower exponential dependence on the uniformity r.

Contribution

It introduces a new proof method for equivalence of testability notions and provides the first explicit upper bounds on sample complexity for hypergraph parameters.

Findings

Sample complexity bounds are exponential towers in r.

New bounds for the r-cut norm of sampled hypergraphs.

Applications to restricted hypergraph parameter classes.

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.