Property Testing in Bounded Degree Hypergraphs

We extend the study of property testing on hypergraphs initiated by Czumaj and Sohler (Theoretical Computer Science, 2005). Provided oracle access to a hypergraph, our goal is to distinguish between the case it has a certain property and the case it is "far" from having this property. Here, we assume that hypergraphs are represented by bounded-length incidence lists and we measure distances between them as a fraction of the maximum possible number of hyperedges. This contrasts with previous work where representations were given by adjacency matrices and distances by fractions of all possible vertex tuples. Thus, while the previous model is better for studying dense hypergraphs, ours is more effective for testing those of bounded-degree. In particular, our model can be seen as an extension to hypergraphs of the graph testing framework introduced by Goldreich and Ron (Algorithmica, 2002). In this framework, we analyse the query complexity of three fundamental hypergraph properties: colorability, $k$-partiteness, and independence number. We show that $k$-partiteness within families of $k$-uniform $n$-vertex hypergraphs of bounded treewidth is strongly testable. Our algorithm always accepts when the hypergraph is $k$-partite, and rejects with high probability if it is $\varepsilon$-far from $k$-partiteness. 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, as an independently interesting combinatorial question, 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.