Property testing and its connection to learning and approximation

The authors study the question of determining whether an unknown function has a particular property or is ε-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, they establish some connections between property testing and problems in learning theory. Next, they focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being k-colorable or having a ρ-clique (clique of density ρ w.r.t. the vertex set). The graph property testing algorithms are probabilistic and make assertions which are correct with high probability utilizing only poly(1/ε) edge-queries into the graph, where ε is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which correspond to the property being tested, if it holds for the input graph