Testing odd-cycle-freeness in Boolean functions

Call a function f : F 2 → {0, 1} odd-cycle-free if there are no x1, . . . , xk ∈ Fn2 with k an odd integer such that f(x1) = · · · = f(xk) = 1 and x1 + · · · + xk = 0. We show that one can distinguish odd-cycle-free functions from those ǫ-far from being odd-cycle-free by making poly(1/ǫ) queries to an evaluation oracle. To obtain this result, we use connections between basic Fourier analysis and spectral graph theory to show that one can reduce testing odd-cyclefreeness of Boolean functions to testing bipartiteness of dense graphs. Our work forms part of a recent sequence of works that shows connections between testability of properties of Boolean functions and of graph properties. We also prove that there is a canonical tester for odd-cycle-freeness making poly(1/ǫ) queries, meaning that the testing algorithm operates by picking a random linear subspace of dimension O(log 1/ǫ) and then checking if the restriction of the function to the subspace is odd-cycle-free or not. The test is analyzed by studying the effect of random subspace restriction on the Fourier coefficients of a function. Our work implies that testing odd-cycle-freeness using a canonical tester instead of an arbitrary tester incurs no more than a polynomial blowup in the query complexity. The question of whether a canonical tester with polynomial blowup exists for all linear-invariant properties remains an open problem.