Features and Limitations
The main efficiency parameter of a property testing algorithm is its query complexity, which is the maximum number of input symbols inspected over all inputs of a given length (and all random choices made by the algorithm). One is interested in designing algorithms whose query complexity is as small as possible. In many cases the running time of property testing algorithms is sublinear in the instance length. Typically, the goal is first to make the query complexity as small as possible as a function of the instance size n, and then study the dependency on the proximity parameter ε.
Unlike other complexity-theoretic settings, the asymptotic query complexity of property testing algorithms is affected dramatically by the representation of instances. For example, when ε = 0.01, the problem of testing bipartiteness of dense graphs (which are represented by their adjacency matrix) admits an algorithm of constant query complexity. In contrast, sparse graphs on n vertices (which are represented by their adjacency list) require property testing algorithms of query complexity .
The query complexity of property testing algorithms grows as the proximity parameter ε becomes smaller for all non-trivial properties. This dependence on ε is necessary as a change of fewer than ε symbols in the input cannot be detected with constant probability using fewer than O(1/ε) queries. Many interesting properties of dense graphs can be tested using query complexity that depends only on ε and not on the graph size n. However, the query complexity can grow enormously fast as a function of ε. For example, for a long time the best known algorithm for testing if a graph does not contain any triangle had a query complexity which is a tower function of poly(1/ε), and only in 2010 this has been improved to a tower function of log(1/ε). One of the reasons for this enormous growth in bounds is that many of the positive results for property testing of graphs are established using the Szemerédi regularity lemma, which also has tower-type bounds in its conclusions.
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