Testing Monotonicity over Graph Products

We consider the problem of monotonicity testing over graph products. Monotonicity testing is one of the central problems studied in the field of property testing. We present a testing approach that enables us to use known monotonicity testers for given graphs C 1 , G 2 , to test monotonicity over their product G 1 x G 2 . Such approach has been previously used in the special case of monotonicity testing over [n] d for a limited type of testers; however, we show that it can be applied to allow modular design of testers in many interesting cases: this approach works whenever the functions are boolean, and also in certain cases for functions with general range. We demonstrate the usefulness of our results by showing how a careful use of this approach improves the query complexity of known testers. Specifically, based on our results, we provide a new analysis for the known tester for [n] d which significantly improves its query complexity analysis in the low-dimensional case. For example, when d = O(1), we reduce the best known query complexity from O(log 2 n/∈) to O(log n/∈).

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