Lower Bounds for Testing Triangle-freeness in Boolean Functions

Let f(1), f(2), f(3) : F-2(n) --> {0, 1} be three Boolean functions We say a triple (x, y, x + y) is a triangle in the function-triple (f(1),f(2),f(3)) if f(1)(x) = f(2)(y) = f(3)(x + y) = 1 (f(1),f(2),f(3)) is said to be triangle-free if there is no triangle in the triple The distance between a function-triple and triangle-freeness is the minimum fraction of function values one needs to modify in order to make the function-triple triangle-free. A canonical tester for triangle-freeness repeatedly picks x and y uniformly and independently at random and checks if f(1)(x) = f(2)(y) = f(3)(x y) = 1 Based on an algebraic regularity lemma, Green showed that the number of queries for the canonical testing algorithm is upper-bounded by a tower of 2's whose height is polynomial in 1/epsilon. A trivial query complexity lower bound of Omega(1/epsilon) is straightforward to show. In this paper, we give the first non-trivial query complexity lower bound for testing triangle-freeness in Boolean functions. We show that, for every small enough c there exists an integer n(0)(epsilon) such that for all n >= n(0) there exists a function-triple f(1), f(2), f(3) F-2(n) --> {0,1} depending on all the n variables which is E-far from being triangle-free and requires (1/epsilon)(4 847) queries for the canonical tester. For the single function case that f(1) = f(2) = f(3), we obtain a weaker lower bound of (1/epsilon)(3 409) We also show that the query complexity of any general (possibly adaptive) one-sided tester for triangle-freeness is at least square-root of the query complexity of the corresponding canonical tester. Consequently, this yields (1/epsilon)(2 423) and (1/epsilon)(1 704) query complexity lower bounds for multi-function and single-function triangle-freeness respectively, with respect to general one-sided testers.