Domain Reduction for Monotonicity Testing: A o(d) Tester for Boolean Functions on Hypergrids
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Testing monotonicity of Boolean functions over the hypergrid, f:[n]^d →{0,1}, is a classic problem in property testing. When the range is real-valued, there are Θ(d n)-query testers and this is tight. In contrast, the Boolean range qualitatively differs in two ways: (1) Independence of n: There are testers with query complexity independent of n [Dodis et al. (RANDOM 1999); Berman et al. (STOC 2014)], with linear dependence on d. (2) Sublinear in d: For the n=2 hypercube case, there are testers with o(d) query complexity [Chakrabarty, Seshadhri (STOC 2013); Khot et al. (FOCS 2015)]. It was open whether one could obtain both properties simultaneously. This paper answers this question in the affirmative. We describe a Õ(d^5/6)-query monotonicity tester for f:[n]^d →{0,1}. Our main technical result is a domain reduction theorem for monotonicity. For any function f, let ϵ_f be its distance to monotonicity. Consider the restriction f̂ of the function on a random [k]^d sub-hypergrid of the original domain. We show that for k = poly(d/ϵ), the expected distance of the restriction E[ϵ_f̂] = Ω(ϵ_f). Therefore, for monotonicity testing in d dimensions, we can restrict to testing over [n]^d, where n = poly(d/ϵ). Our result follows by applying the d^5/6·poly(1/ϵ, n, d)-query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018).
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