We describe a @math -query monotonicity tester for Boolean functions @math on the @math -hypergrid. This is the first @math monotonicity tester with query complexity independent of @math . Motivated by this independence of @math , we initiate the study of monotonicity testing of measurable Boolean functions @math over the continuous domain, where the distance is measured with respect to a product distribution over @math . We give a @math -query monotonicity tester for such functions. Our main technical result is a domain reduction theorem for monotonicity. For any function @math , let @math be its distance to monotonicity. Consider the restriction @math of the function on a random @math sub-hypergrid of the original domain. We show that for @math , the expected distance of the restriction is @math . Previously, such a result was only known for @math (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014). Our result for testing Boolean functions over @math then follows by applying the @math -query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018). To obtain the result for testing Boolean functions over @math , we use standard measure theoretic tools to reduce monotonicity testing of a measurable function @math to monotonicity testing of a discretized version of @math over a hypergrid domain @math for large, but finite, @math (that may depend on @math ). The independence of @math in the hypergrid tester is crucial to getting the final tester over @math .
We give a short summary of Boolean monotonicity testing over the hypercube. The problem was introduced by @cite_2 (refer to Raskhodnikova's thesis @cite_14 ), with an @math -query tester. The first improvement over that bound was the @math tester of Chakrabarty and Seshadhri @cite_6 , achieved via a directed analogue of Margulis' isoperimetric theorem. Chen-Servedio-Tan improved the analysis to get an @math bound @cite_27 . A breakthrough result of Khot-Minzer-Safra gave an @math tester @cite_29 . All these testers are non-adaptive and one-sided. had proved a (nearly) matching lower bound of @math for this case @cite_26 . The first polynomial two-sided lower bound was given by Chen-Servedio-Tan, subsequently improved to @math by @cite_1 . The first polynomial lower bound of @math for adaptive testers was given recently by Belovs-Blais @cite_15 , and was improved to @math by Chen-Waingarten-Xie @cite_31 .
@cite_31 @cite_14 @cite_26 @cite_29 @cite_1 @cite_6 @cite_27 @cite_2 @cite_15
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We prove a lower bound of Ω(n1 3) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f: 0,1 n→ 0,1 is monotone versus far from monotone. This improves the recent lower bound of Ω(n1 4) for the same problem by Belovs and Blais (STOC'16). Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity we prove a lower bound of Ω(√n) for two-sided, adaptive algorithms and a lower bound of Ω(n) for one-sided, non-adaptive algorithms for testing unateness, a natural generalization of monotonicity. The latter matches the linear upper bounds by Khot and Shinkar (RANDOM'16) and by Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri (2017). We study monotonicity testing of Boolean functions over the hypergrid [n]d and design a non-adaptive tester with 1-sided error whose query complexity is O(d5 6) · poly(log n, 1 e). Previous to our work, the best known testers had query complexity linear in d but independent of n. We improve upon these testers as long as n = 2do(1). To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure. We consider the problem of testing whether an unknown Boolean function f : -- 1, 1 n a#x21C6; -- 1, 1 is monotone versus aepsi;-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied problem. Lower bound: We prove an a#x03A9;(n1 5) lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of a#x03A9;(log n) due to [1]. We show that the same lower bound holds for monotonicity testing of Boolean-valued functions over hypergrid domains 1,,m n for all m age; 2. Upper bound: We present an O(n5 6) poly(1 aepsi;)-query algorithm that tests whether an unknown Boolean function f is monotone versus aepsi;-far from monotone. Our algorithm, which is non-adaptive and makes one-sided error, is a modified version of the algorithm of Chakrabarty and Seshadhri[2], which makes O(n7 8) poly(1 aepsi;) queries. We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes @math nonadaptive queries to a function @math , always accepts a monotone function, and rejects a function that is @math -far from being monotone with constant probability.