Testing Graph Properties with the Container Method

We establish nearly optimal sample complexity bounds for testing the $\rho$-clique property in the dense graph model. Specifically, we show that it is possible to distinguish graphs on $n$ vertices that have a $\rho n$-clique from graphs for which at least $\epsilon n^2$ edges must be added to form a $\rho n$-clique by sampling and inspecting a random subgraph on only $\tilde{O}(\rho^3/\epsilon^2)$ vertices. We also establish new sample complexity bounds for $\epsilon$-testing $k$-colorability. In this case, we show that a sampled subgraph on $\tilde{O}(k/\epsilon)$ vertices suffices to distinguish $k$-colorable graphs from those for which any $k$-coloring of the vertices causes at least $\epsilon n^2$ edges to be monochromatic. The new bounds for testing the $\rho$-clique and $k$-colorability properties are both obtained via new extensions of the graph container method. This method has been an effective tool for tackling various problems in graph theory and combinatorics. Our results demonstrate that it is also a powerful tool for the analysis of property testing algorithms.