
arXiv: 2502.18382
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.
Added references; clarified historical account (abstract and introduction)
FOS: Computer and information sciences, Computer Science - Computational Complexity, Discrete Mathematics (cs.DM), Computational Complexity (cs.CC), Computer Science - Discrete Mathematics
FOS: Computer and information sciences, Computer Science - Computational Complexity, Discrete Mathematics (cs.DM), Computational Complexity (cs.CC), Computer Science - Discrete Mathematics
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