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

Authors Elena Grigorescu, Akash Kumar, Karl Wimmer

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Author Details

Elena Grigorescu
  • Purdue University, West Lafayette, IN, USA,
Akash Kumar
  • Purdue University, West Lafayette, IN, USA,
Karl Wimmer
  • Duquesne University, Pittsburgh, PA, USA,

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Elena Grigorescu, Akash Kumar, and Karl Wimmer. Flipping out with Many Flips: Hardness of Testing k-Monotonicity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 40:1-40:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Lower bounds and information complexity
  • Property Testing
  • Boolean Functions
  • k-Monotonicity
  • Lower Bounds


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  1. Nir Ailon and Bernard Chazelle. Information theory in property testing and monotonicity testing in higher dimension. Inf. Comput., 204(11):1704-1717, 2006. Google Scholar
  2. Maria-Florina Balcan, Eric Blais, Avrim Blum, and Liu Yang. Active property testing. In FOCS, pages 21-30. IEEE Computer Society, 2012. Google Scholar
  3. Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri. Optimal unateness testers for real-valued functions: Adaptivity helps. CoRR, abs/1703.05199, 2017. Google Scholar
  4. Tŭgkan Batu, Ronitt Rubinfeld, and Patrick White. Fast approximate PCPs for multidimensional bin-packing problems. Inf. Comput., 196(1):42-56, 2005. Google Scholar
  5. Aleksandrs Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In STOC, pages 1021-1032. ACM, 2016. Google Scholar
  6. Omri Ben-Eliezer and Clément L. Canonne. Improved bounds for testing forbidden order patterns. In SODA 2018, New Orleans, LA, USA, January 7-10, 2018, 2018. Google Scholar
  7. Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In STOC, pages 164-173. ACM, 2014. Google Scholar
  8. Arnab Bhattacharyya, Victor Chen, Madhu Sudan, and Ning Xie. Testing linear-invariant non-linear properties. Theory of Computing, 7(1):75-99, 2011. Google Scholar
  9. Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett. Every locally characterized affine-invariant property is testable. In STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 429-436, 2013. Google Scholar
  10. Arnab Bhattacharyya, Eldar Fischer, and Shachar Lovett. Testing low complexity affine-invariant properties. In SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1337-1355, 2013. Google Scholar
  11. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. In SODA, pages 932-941. SIAM, 2009. Google Scholar
  12. Arnab Bhattacharyya, Elena Grigorescu, and Asaf Shapira. A unified framework for testing linear-invariant properties. Random Struct. Algorithms, 46(2):232-260, 2015. Google Scholar
  13. Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. Computational Complexity, 21(2):311-358, 2012. Google Scholar
  14. Eric Blais, Clément L. Canonne, Igor Carboni Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning circuits with few negations. In APPROX-RANDOM, volume 40 of LIPIcs, pages 512-527. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  15. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci., 47(3):549-595, 1993. Google Scholar
  16. Jop Briët, Sourav Chakraborty, David García-Soriano, and Arie Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012. Google Scholar
  17. Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-monotonicity. In ITCS, page (to appear), 2017. Google Scholar
  18. Clément L. Canonne and Ronitt Rubinfeld. Testing probability distributions underlying aggregated data. In ICALP (1), volume 8572 of Lecture Notes in Computer Science, pages 283-295. Springer, 2014. Google Scholar
  19. Deeparnab Chakrabarty and C. Seshadhri. An o(n) monotonicity tester for boolean functions over the hypercube. In STOC, pages 411-418. ACM, 2013. Journal version as [22]. Google Scholar
  20. Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In STOC, pages 419-428, 2013. Google Scholar
  21. Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014. Google Scholar
  22. Deeparnab Chakrabarty and C. Seshadhri. An o(n) Monotonicity Tester for Boolean Functions over the Hypercube. SIAM J. Comput., 45(2):461-472, 2016. Google Scholar
  23. Deeparnab Chakrabarty and C. Seshadhri. A dollarbackslashwidetildeO(n)dollar non-adaptive tester for unateness. CoRR, 2016. Google Scholar
  24. Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean function monotonicity testing requires (almost) n^1/2 non-adaptive queries. In STOC, pages 519-528. ACM, 2015. Google Scholar
  25. Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In FOCS, pages 286-295. IEEE Computer Society, 2014. Google Scholar
  26. Xi Chen, Erik Waingarten, and Jinyu Xie. Beyond talagrand functions: New lower bounds for testing monotonicity and unateness. CoRR, abs/1702.06997, 2017. Google Scholar
  27. Krishnamoorthy Dinesh and Jayalal Sarma. Alternation, sparsity and sensitivity : Bounds and exponential gaps. In CALDAM 2018. Springer, 2018. Google Scholar
  28. Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In RANDOM-APPROX, volume 1671 of Lecture Notes in Computer Science, pages 97-108. Springer, 1999. Google Scholar
  29. Benjamin Doerr. Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2011. Google Scholar
  30. Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. J. Comput. Syst. Sci., 60(3):717-751, 2000. Google Scholar
  31. Shahar Fattal and Dana Ron. Approximating the distance to monotonicity in high dimensions. ACM Trans. Algorithms, 6(3), 2010. Google Scholar
  32. Eldar Fischer. On the strength of comparisons in property testing. Inf. Comput., 189(1):107-116, 2004. Google Scholar
  33. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In STOC 2002, Montréal, Québec, Canada, pages 474-483, 2002. Google Scholar
  34. Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301-337, 2000. Google Scholar
  35. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. Google Scholar
  36. Ben Green. A Szemerédi-type regularity lemma in abelian groups. Geometric and Functional Analysis, 15(2):340-376, 2005. Google Scholar
  37. Siyao Guo and Ilan Komargodski. Negation-limited formulas. In APPROX-RANDOM, volume 40 of LIPIcs, pages 850-866. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  38. Siyao Guo, Tal Malkin, Igor Carboni Oliveira, and Alon Rosen. The power of negations in cryptography. In TCC (1), volume 9014 of Lecture Notes in Computer Science, pages 36-65. Springer, 2015. Google Scholar
  39. Shirley Halevy and Eyal Kushilevitz. Testing monotonicity over graph products. Random Struct. Algorithms, 33(1):44-67, 2008. Google Scholar
  40. Stasys Jukna. Boolean Function Complexity. Springer, 2012. Google Scholar
  41. Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 403-412. ACM, 2008. URL:
  42. Michael J. Kearns and Dana Ron. Testing problems with sublearning sample complexity. J. Comput. Syst. Sci., 61(3):428-456, 2000. Google Scholar
  43. Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and Boolean isoperimetric type theorems. In FOCS, pages 52-58. IEEE Computer Society, 2015. Google Scholar
  44. Subhash Khot and Igor Shinkar. An asciitildeo(n) queries adaptive tester for unateness. In APPROX/RANDOM 2016, Paris, France, pages 37:1-37:7, 2016. Google Scholar
  45. Pravesh Kothari, Amir Nayyeri, Ryan O'Donnell, and Chenggang Wu. Testing surface area. In SODA, pages 1204-1214. SIAM, 2014. Google Scholar
  46. Daniel Král', Oriol Serra, and Lluís Vena. A removal lemma for systems of linear equations over finite fields. Israel Journal of Mathematics, pages 1-15, 2011. URL:
  47. Chengyu Lin and Shengyu Zhang. Sensitivity conjecture and log-rank conjecture for functions with small alternating numbers. CoRR, abs/1602.06627, 2016. Google Scholar
  48. A. A. Markov. On the inversion complexity of systems of functions. Doklady Akademii Nauk SSSR, 116:917-919, 1957. English translation in [49]. Google Scholar
  49. A. A. Markov. On the inversion complexity of a system of functions. Journal of the ACM, 5(4):331-334, October 1958. Google Scholar
  50. Joe Neeman. Testing surface area with arbitrary accuracy. In STOC, pages 393-397. ACM, 2014. Google Scholar
  51. Ilan Newman, Yuri Rabinovich, Deepak Rajendraprasad, and Christian Sohler. Testing for forbidden order patterns in an array. In SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1582-1597, 2017. Google Scholar
  52. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012-1042, 2006. URL:
  53. Benjamin Rossman. Correlation bounds against monotone NC¹. In Conference on Computational Complexity (CCC), 2015. Google Scholar
  54. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM-J-COMPUT, 25(2):252-271, 1996. Google Scholar
  55. Asaf Shapira. Green’s conjecture and testing linear invariant properties. In Property Testing - Current Research and Surveys [outgrow of a workshop at the Institute for Computer Science (ITCS) at Tsinghua University, January 2010], pages 355-358, 2010. Google Scholar
  56. Yuichi Yoshida. A characterization of locally testable affine-invariant properties via decomposition theorems. In STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 154-163, 2014. Google Scholar
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