A Characterization of Testable Hypergraph Properties

2017-07-11

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This paper addresses the combinatorial characterization of testable properties of $k$-uniform hypergraphs: namely, which properties $P$ admit constant-query testers that distinguish, with probability at least $2/3$, hypergraphs satisfying $P$ from those that are $varepsilon$-far from $P$. The authors provide the first complete combinatorial characterization of hypergraph testability, establishing its equivalence to the hypergraph regularity lemma, hypergraph limits (a.k.a. graphons for higher-order structures), and representability. Their main result proves that, for hypergraphs, testability, learnability, and approximability are strictly equivalent—resolving a fundamental question in property testing. This unifies and substantially generalizes classical results for graphs, extending the theoretical foundations of property testing to higher-order combinatorial structures. The equivalence yields a robust, unified framework for analyzing and designing efficient testers for complex, high-dimensional relational data, with implications for extremal combinatorics, algorithmic learning theory, and large-scale data analysis.

Problem

• Characterize testable properties of k-uniform hypergraphs
• Distinguish hypergraphs satisfying P vs far from P
• Extend graph testability results to k-graphs

Innovation

• Combinatorial characterization of testable hypergraph properties
• Randomized algorithm with bounded edge queries
• Random subhypergraph sampling for proof