On Being Far from Far and on Dual Problems in Property Testing

This work studies a new type of problems in property testing, called dual problems. For a set Π in a metric space and δ > 0, denote by Fδ(Π) the set of elements that are δ-far from Π. Then, in property testing, a δ-tester for Π is required to accept inputs from Π and reject inputs from Fδ(Π). A natural dual problem is the problem of δ-testing the set of “no” instances, that is Fδ(Π): A δ-tester for Fδ(Π) needs to accept inputs from Fδ(Π) and reject inputs that are δ-far from Fδ(Π); that is, it rejects inputs from Fδ(Fδ(Π)). When Π = Fδ(Fδ(Π)) the dual problem is essentially equivalent to the original one, but this equality does not hold in general. Many dual problems constitute appealing testing problems that are interesting by themselves. In this work we study sets of the form Fδ(Fδ(Π)), and apply this study to investigate several natural dual problems. In particular, we derive lower bounds and upper bounds on the query complexity of several classes of natural dual problems: These include dual problems of properties of functions (e.g., testing error-correcting codes and testing monotone functions), of properties of distributions (e.g., testing equivalence to a known distribution), and of various graph properties in the dense graph model and in the bounded-degree model. We also show that testing any dual problem with one-sided error is either trivial or requires a linear number of queries.