License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.40
URN: urn:nbn:de:0030-drops-94448
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9444/
Go to the corresponding LIPIcs Volume Portal


Grigorescu, Elena ; Kumar, Akash ; Wimmer, Karl

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

pdf-format:
LIPIcs-APPROX-RANDOM-2018-40.pdf (0.5 MB)


Abstract

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.

BibTeX - Entry

@InProceedings{grigorescu_et_al:LIPIcs:2018:9444,
  author =	{Elena Grigorescu and Akash Kumar and Karl Wimmer},
  title =	{{Flipping out with Many Flips: Hardness of Testing k-Monotonicity}},
  booktitle =	{Approximation, Randomization, and Combinatorial  Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9444},
  URN =		{urn:nbn:de:0030-drops-94448},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.40},
  annote =	{Keywords: Property Testing, Boolean Functions, k-Monotonicity, Lower Bounds}
}

Keywords: Property Testing, Boolean Functions, k-Monotonicity, Lower Bounds
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 13.08.2018


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI