Testing Linear Inequalities of Subgraph Statistics

Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property ? and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs ℱ, the property of being induced ℱ-free (i.e. not containing an induced copy of any F ∈ ℱ) is testable. It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs ℱ, and any linear inequality involving the densities of the graphs F ∈ ℱ in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing. The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a pseudo-random graph.