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A characterization of testable hypergraph properties

Felix Joos, Jaehoon Kim, Daniela Kühn, Deryk Osthus

TL;DR

The paper characterizes testable properties of dense $k$-uniform hypergraphs by proving that a $k$-graph property is testable if and only if it is regular reducible, extending the graph-case result of AFNS09 to hypergraphs. The approach combines strong hypergraph regularity with a regular-approximation framework and a random subhypergraph sampling tool from a companion paper to reduce testing to a bounded collection of regularity instances $R=(\varepsilon,\mathbf{a},d_{\mathbf{a},k})$, and to relate sample-based evidence to global structure via induced counting lemmas. A key outcome is the equivalence between testability and estimability for hypergraph properties, enabling explicit estimators built from a finite set of regularity instances. The framework yields concrete applications, including testing the injective homomorphism density and the maximum $\,\ell$-way cut, demonstrating the practical reach of the characterization despite the typically tower-type quantitative bounds. Overall, the results unify property testing, hypergraph regularity, and graph limits into a cohesive theory for higher-order combinatorial structures with broad implications for combinatorics and theoretical computer science.

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.

A characterization of testable hypergraph properties

TL;DR

The paper characterizes testable properties of dense -uniform hypergraphs by proving that a -graph property is testable if and only if it is regular reducible, extending the graph-case result of AFNS09 to hypergraphs. The approach combines strong hypergraph regularity with a regular-approximation framework and a random subhypergraph sampling tool from a companion paper to reduce testing to a bounded collection of regularity instances , and to relate sample-based evidence to global structure via induced counting lemmas. A key outcome is the equivalence between testability and estimability for hypergraph properties, enabling explicit estimators built from a finite set of regularity instances. The framework yields concrete applications, including testing the injective homomorphism density and the maximum -way cut, demonstrating the practical reach of the characterization despite the typically tower-type quantitative bounds. Overall, the results unify property testing, hypergraph regularity, and graph limits into a cohesive theory for higher-order combinatorial structures with broad implications for combinatorics and theoretical computer science.

Abstract

We provide a combinatorial characterization of all testable properties of -uniform hypergraphs (-graphs for short). Here, a -graph property is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability between -graphs that satisfy and those that are far from satisfying . For the -graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the -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 -graph setting. Our proof relies on a random subhypergraph sampling result proved in a companion paper.

Paper Structure

This paper contains 29 sections, 38 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof sketch
  3. Testable properties are regular reducible
  4. Satisfying a regularity instance is testable
  5. The final step
  6. Concepts and tools
  7. Hypergraphs
  8. Probabilistic tools
  9. Hypergraph regularity
  10. Partitions of hypergraphs and the regular approximation lemma
  11. The address space
  12. Basic properties of the address space
  13. Constructing families of partitions using the address space
  14. Density functions of address spaces
  15. Regularity instances
  16. ...and 14 more sections

Key Result

Theorem 1.3

Suppose $k\in\mathbb N\setminus\{1\}$ and suppose $\mathbf{P}$ is a $k$-graph property. Then the following three statements are equivalent:

Figures (1)

  • Figure 1: The proof strategy for Theorem \ref{['thm: estimable']} (Case 1).

Theorems & Definitions (63)

  • Definition 1.1: Testability
  • Definition 1.2: Estimability
  • Theorem 1.3
  • Proposition 3.1
  • Lemma 3.2: See JLR00
  • Lemma 3.3
  • Definition 3.4: Family of partitions
  • Proposition 3.5
  • Definition 3.6: Equitable family of partitions
  • Definition 3.7: Perfectly regular
  • ...and 53 more