A characterization of testable hypergraph properties

TL;DR

This paper characterizes all testable properties of k-uniform hypergraphs using combinatorial methods, extending graph testability results to hypergraphs and contrasting with local repairability limitations.

Contribution

It provides a combinatorial characterization of testable properties for k-uniform hypergraphs, generalizing previous graph results and highlighting differences with local repairability.

Findings

Characterization of all testable hypergraph properties

Extension of graph testability results to hypergraphs

Contrast with local repairability limitations in hypergraphs

Abstract

We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability $2/3$ between $k$-graphs that satisfy $P$ and those that are far from satisfying $P$. For the $2$-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the $k$-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the $3$-graph setting. Our proof relies on a random subhypergraph sampling result proved in a companion paper.