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Property Testing in Bounded Degree Hypergraphs

Hugo Aaronson, Gaia Carenini, Atreyi Chanda

TL;DR

This work generalizes the bounded-degree property-testing framework from graphs to $k$-uniform hypergraphs, and analyzes the sublinear query complexity for three core properties: colorability, $k$-partiteness, and independence number. It introduces a local, gap-preserving reduction $\ ho_{3\text{-col.}}$ from $(3,d)$-SAT to 3-uniform hypergraph colorability, using expander-based gadgets to enforce variable consistency and clause satisfaction, and proves $\ ilde{\Omega}(n)$-hardness for testing colorability and partiteness in bounded-degree hypergraphs. It further shows that while $k$-partiteness is maximally hard in general, it is strongly testable for hypergraphs with bounded treewidth, via a treewidth-preserving reduction to graph colorability. The paper also establishes hardness for testing independence number, and extends the lower bounds to two-sided testers while providing explicit hard instances through CSP-based constructions. These results highlight a sharp separation between dense and bounded-degree hypergraph models and extend the toolkit for sublinear hypergraph property testing.

Abstract

We extend the bounded degree graph model for property testing introduced by Goldreich and Ron (Algorithmica, 2002) to hypergraphs. In this framework, we analyse the query complexity of three fundamental hypergraph properties: colorability, $k$-partiteness, and independence number. We present a randomized algorithm for testing $k$-partiteness within families of $k$-uniform $n$-vertex hypergraphs of bounded treewidth whose query complexity does not depend on $n$. In addition, we prove optimal lower bounds of $Ω(n)$ on the query complexity of testing algorithms for $k$-colorability, $k$-partiteness, and independence number in $k$-uniform $n$-vertex hypergraphs of bounded degree. For each of these properties, we consider the problem of explicitly constructing $k$-uniform hypergraphs of bounded degree that differ in $Θ(n)$ hyperedges from any hypergraph satisfying the property, but where violations of the latter cannot be detected in any neighborhood of $o(n)$ vertices.

Property Testing in Bounded Degree Hypergraphs

TL;DR

This work generalizes the bounded-degree property-testing framework from graphs to -uniform hypergraphs, and analyzes the sublinear query complexity for three core properties: colorability, -partiteness, and independence number. It introduces a local, gap-preserving reduction from -SAT to 3-uniform hypergraph colorability, using expander-based gadgets to enforce variable consistency and clause satisfaction, and proves -hardness for testing colorability and partiteness in bounded-degree hypergraphs. It further shows that while -partiteness is maximally hard in general, it is strongly testable for hypergraphs with bounded treewidth, via a treewidth-preserving reduction to graph colorability. The paper also establishes hardness for testing independence number, and extends the lower bounds to two-sided testers while providing explicit hard instances through CSP-based constructions. These results highlight a sharp separation between dense and bounded-degree hypergraph models and extend the toolkit for sublinear hypergraph property testing.

Abstract

We extend the bounded degree graph model for property testing introduced by Goldreich and Ron (Algorithmica, 2002) to hypergraphs. In this framework, we analyse the query complexity of three fundamental hypergraph properties: colorability, -partiteness, and independence number. We present a randomized algorithm for testing -partiteness within families of -uniform -vertex hypergraphs of bounded treewidth whose query complexity does not depend on . In addition, we prove optimal lower bounds of on the query complexity of testing algorithms for -colorability, -partiteness, and independence number in -uniform -vertex hypergraphs of bounded degree. For each of these properties, we consider the problem of explicitly constructing -uniform hypergraphs of bounded degree that differ in hyperedges from any hypergraph satisfying the property, but where violations of the latter cannot be detected in any neighborhood of vertices.

Paper Structure

This paper contains 20 sections, 21 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Techniques
  4. Structure of the Paper
  5. Hardness of Testing Hypergraph Colorability
  6. Description of the algorithm $\rho_{\text{3-col.}}$
  7. Overview
  8. Construction
  9. Correctness of $\rho_{\text{3-col.}}$
  10. Testing Hypergraph $k$-Partiteness
  11. Hardness of Testing Hypergraph $k$-Partiteness
  12. Testability of $k$-Partiteness in Bounded Treewidth Hypergraphs
  13. Hardness of Testing Hypergraph Independence Number
  14. Lower Bounds for Two-Sided Error Testers and Explicit Constructions
  15. Background
  16. ...and 5 more sections

Key Result

Theorem 1.1

For every $k\geq 3$, there are positive constants $d_{k\text{-col.}}, \varepsilon_{k\text{-col.}}$, such that the query complexity of testing $k$-colorability of $k$-uniform $n$-vertex hypergraphs of bounded degree $d_{k\text{-col.}}$ is $\Omega(n)$.

Figures (1)

  • Figure 1: Schematic representation of the clause gadget for the clause $(x_1\vee x_2\vee x_3)$.

Theorems & Definitions (30)

  • Theorem 1.1: Hardness of testing hypergraph $k$-colorability
  • Corollary 1.2: Dense vs. bounded degree model for hypergraphs, informal
  • Theorem 1.3: Hardness of testing hypergraph $k$-partiteness
  • Theorem 1.4: Strong testability of $k$-partiteness in bounded treewidth hypergraph
  • Theorem 1.5: Hardness of testing independence number
  • Theorem 2.1: Hardness of testing hypergraph $3$-colorability
  • Theorem 2.2: Hardness of testing $(3,d_{\text{SAT}})$-SAT BOT02
  • Lemma 2.3: Correctness of $\rho_{\text{3-col.}}$
  • proof
  • Theorem 3.1: Hardness of testing hypergraph $3$-partiteness
  • ...and 20 more