2026-10-15T19:48:20Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9444 2024-03-06T09:44:25Z ddc:004 open_access

Flipping out with Many Flips: Hardness of Testing k-Monotonicity Grigorescu, Elena Kumar, Akash Wimmer, Karl Property Testing Boolean Functions k-Monotonicity Lower Bounds A function f:{0,1}^n - > {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. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2017) 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 et al. (JCSS, 72(6), 2006). In this process we resolve a problem left open in previous work. Specifically, our results include the following. 1) Testing 2-monotonicity on the hypercube non-adaptively 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 O~(sqrt{n}) queries (Khot et al. (FOCS 2015)). 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. 2) On the hypercube [n]^d domain, there exists a testing algorithm that makes a constant number of queries and distinguishes functions that are k-monotone from functions that are far from being O(kd^2) -monotone. Such a dependency is likely necessary, given the lower bound above for the hypercube. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Elena Grigorescu and Akash Kumar and Karl Wimmer 2018 Is Part Of LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2018.40 urn:nbn:de:0030-drops-94448 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.40 eng https://creativecommons.org/licenses/by/3.0/legalcode