Property testing of massively parametrized problems - A survey

We survey property testing results for the so called 'massively parametrized' model (or problems). The massively-parametrized framework studies problems such as: Given a graph G = (V,E), consider the property BI(G) containing all subgraphs of G that are bipartite. Massively parametrized Properties, such as BI(G), are defined by two ingredients: a 'general' property of inputs (e.g, 'being bipartite' for the example above), and an underlying given structure (the graph G for the property BI(G)). Then the property is the restriction of the general property to inputs that are associated with the given structure. In such situation, keeping the general property fixed and varying the underlying structure, the property and the corresponding tests might vary significantly in their structure. The focus of the study in this framework is how, for a fixed general property, the testing problem changes while changing the underlying structure.