Polynomial property testing

Property testers are fast, randomized “election polling”-type algorithms that determine if an input (e.g., graph or hypergraph) has a certain property or is $\varepsilon$-far from the property. It is known that many properties can be tested with query complexity that depends only on the error parameter $\varepsilon$ (and not on the size of the input), but the current bounds on the query complexity grow extremely quickly as a function of $\frac{1}{\varepsilon }$. Which properties can be tested efficiently, i.e., with $\text{poly}(1/\varepsilon )$ queries? This survey presents the state of knowledge on this general question, focusing on the dense graph model of property testing. Several key open problems are presented.