Property Testing in Bounded Degree Hypergraphs

TL;DR

This paper extends property testing to hypergraphs, analyzing query complexities for key properties, and provides algorithms and lower bounds demonstrating the limits of testing in bounded degree hypergraphs.

Contribution

It introduces a hypergraph property testing framework, presents a query-efficient algorithm for k-partiteness, and establishes optimal lower bounds for testing fundamental properties.

Findings

Query complexity for k-partiteness does not depend on n in certain hypergraph families.

Optimal lower bounds of Ω(n) for testing k-colorability, k-partiteness, and independence number.

Construction of hypergraphs that differ significantly from property-satisfying hypergraphs without detectable violations.

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 $\Omega(n)$ 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 $\Theta(n)$ hyperedges from any hypergraph satisfying the property, but where violations of the latter cannot be detected in any neighborhood of $o(n)$ vertices.