On the Testability of Graph Partition Properties

In this work we study the testability of a family of graph partition properties that generalizes a family previously studied by Goldreich, Goldwasser, and Ron (Journal of the ACM, 1998 ). While the family studied by Goldreich, Goldwasser, and Ron includes a variety of natural properties, such as k-colorability and containing a large cut, it does not include other properties of interest, such as split graphs, and more generally (p,q)-colorable graphs. The generalization we consider allows us to impose constraints on the edge-densities within and between parts (relative to the sizes of the parts). We denote the family studied in this work by GPP. We first show that all properties in GPP have a testing algorithm whose query complexity is polynomial in 1/epsilon, where epsilon is the given proximity parameter (and there is no dependence on the size of the graph). As the testing algorithm has two-sided error, we next address the question of which properties in GPP can be tested with one-sided error and query complexity polynomial in 1/epsilon. We answer this question by establishing a characterization result. Namely, we define a subfamily GPP_{0,1} of GPP and show that a property P in GPP is testable by a one-sided error algorithm that has query complexity poly(1/epsilon) if and only if P in GPP_{0,1}.