Flipping Out with Many Flips: Hardness of Testing k-Monotonicity

Elena Grigorescu, Akash Kumar, Karl Wimmer

Abstract: A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone if it flips between 0 and 1 at most $k$ times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. [Innovations in Theoretical Computer Science, Schloss-Dagstuhl--Leibniz-Zentrum für Informatik GmBH, Wadern, Germany, 2017, 29] initiate the study of $k$-monotone functions in the area of property testing, and Newman et al. [SODA, SIAM, Philadelphia, 2017, pp. 1582--1597] study testability of families characterized by freeness from order patterns on real-valued functions over the line $[n]$ domain. We study $k$-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas, Ron, and Rubinfeld [J. Comput. System Sci., 72 (2006), pp. 1012--1042]. In this process we show strong lower bounds on testing $k$-monotonicity. Specifically, we show that testing 2-monotonicity on the hypercube nonadaptively with one-sided error requires an exponential in $\sqrt{n}$ number of queries. This behavior shows a stark contrast with testing (1-)monotonicity, which only needs $\tilde{O}\mleft(\sqrt{n}\mright)$ queries. Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being $n^{.01}$-monotone also requires an exponential number of queries.