Testability of relations between permutations

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple (σ1, . . . , σd) of permutations on {1, . . . , n}, and one wishes to determine whether this tuple satisfies a certain system of relations E, or is far from every tuple that satisfies E. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d = 2 and E consists of the single relation XY = YX, this corresponds to testing whether σ1σ2 = σ2σ1, where σ1σ2 and σ2σ1 denote composition of permutations. We define a collection of graphs, naturally associated with the system E, that encodes all the information relevant to the testability of E. We then prove two theorems that provide criteria for testability and nontestability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testability notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability.