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ABSTRACT: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 G 1 , G 2 , to test monotonicity over their product G 1 × G 2 . Such an approach of reducing monotonicity testing over a graph product to monotonicity testing over the original graphs, has been previously used in the special case of monotonicity testing over [n] d for a limited type of testers; in this article, we show that this approach 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 a 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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We study a model of graph related formulae that we call the Constraint-Graph model. A constraintgraph is a labeled multi-graph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over the variables that label its adjacent edges. A Boolean assignment to the variables satisfies the constraint graph if it satisfies every vertex function. We associate with a constraint-graph G the property of all assignments satisfying G, denoted SAT (G). We show that the above model is quite general. That is, for every property of strings P there exists a property of constraint-graphs P G such that P is testable using q queries iff P G is thus testable. In addition, we present a large family of constraint-graphs for which SAT (G) is testable with constant number of queries. As an implication of this, we infer the testability of some edge coloring problems (e.g. the property of two coloring of the edges in which every node is adjacent to at least one vertex of each color). Another implication is that every property of Boolean strings that can be represented by a Read-twice CNF formula is testable. We note that this is the best possible in terms of the number of occurrences of every variable in a formula.