Document Open Access Logo

On the Sensitivity Complexity of k-Uniform Hypergraph Properties

Authors Qian Li, Xiaoming Sun

PDF
Thumbnail PDF

File

LIPIcs.STACS.2017.51.pdf
  • Filesize: 0.48 MB
  • 12 pages

Document Identifiers

Author Details

Qian Li
Xiaoming Sun

Cite As Get BibTex

Qian Li and Xiaoming Sun. On the Sensitivity Complexity of k-Uniform Hypergraph Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.STACS.2017.51

Abstract

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k.

Subject Classification

Keywords
  • Sensitivity Complexity
  • k-uniform Hypergraph Properties
  • Boolean Function
  • Turan's question

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Andris Ambainis, Mohammad Bavarian, Yihan Gao, Jieming Mao, Xiaoming Sun, and Song Zuo. Tighter relations between sensitivity and other complexity measures. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 101-113, 2014. Google Scholar
  2. Andris Ambainis and Krisjanis Prusis. A tight lower bound on certificate complexity in terms of block sensitivity and sensitivity. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, pages 33-44, 2014. Google Scholar
  3. Andris Ambainis, Krisjanis Prusis, and Jevgenijs Vihrovs. Sensitivity versus certificate complexity of boolean functions. In Computer Science - Theory and Applications - 11th International Computer Science Symposium in Russia, CSR 2016, St. Petersburg, Russia, June 9-13, 2016, Proceedings, pages 16-28, 2016. Google Scholar
  4. Andris Ambainis and Xiaoming Sun. New separation between s(f) and bs(f). Electronic Colloquium on Computational Complexity (ECCC), 18:116, 2011. Google Scholar
  5. Andris Ambainis and Jevgenijs Vihrovs. Size of sets with small sensitivity: A generalization of simon’s lemma. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, Proceedings, pages 122-133, 2015. Google Scholar
  6. László Babai, Anandam Banerjee, Raghav Kulkarni, and Vipul Naik. Evasiveness and the distribution of prime numbers. In 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, March 4-6, 2010, Nancy, France, pages 71-82, 2010. Google Scholar
  7. Mitali Bafna, Satyanarayana V. Lokam, Sébastien Tavenas, and Ameya Velingker. On the sensitivity conjecture for read-k formulas. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 16:1-16:14, 2016. Google Scholar
  8. Shalev Ben-David. Low-sensitivity functions from unambiguous certificates. Electronic Colloquium on Computational Complexity (ECCC), 23:84, 2016. Google Scholar
  9. Stella Biderman, Kevin Cuddy, Ang Li, and Min Jae Song. On the sensitivity of k-uniform hypergraph properties. CoRR, abs/1510.00354, 2015. Google Scholar
  10. Timothy Black. Monotone properties of k-uniform hypergraphs are weakly evasive. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS'15, pages 383-391, 2015. Google Scholar
  11. Meena Boppana. Lattice variant of the sensitivity conjecture. Electronic Colloquium on Computational Complexity (ECCC), 19:89, 2012. Google Scholar
  12. Harry Buhrman and Ronald De Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. Google Scholar
  13. Amit Chakrabarti, Subhash Khot, and Yaoyun Shi. Evasiveness of subgraph containment and related properties. SIAM J. Comput., 31(3):866-875, 2001. Google Scholar
  14. Sourav Chakraborty. On the sensitivity of cyclically-invariant boolean functions. In Proceedings of the 20th Annual IEEE Conference on Computational Complexity, CCC'05, pages 163-167, 2005. Google Scholar
  15. Sourav Chakraborty. Sensitivity, Block Sensitivity and Certificate Complexity of Boolean Functions. Master’s thesis, University of Chicago, 2005. Google Scholar
  16. Stephen Cook and Cynthia Dwork. Bounds on the time for parallel ram’s to compute simple functions. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC'82, pages 231-233, 1982. Google Scholar
  17. Stephen A. Cook, Cynthia Dwork, and Rüdiger Reischuk. Upper and lower time bounds for parallel random access machines without simultaneous writes. SIAM J. Comput., 15(1):87-97, 1986. Google Scholar
  18. Yihan Gao, Jieming Mao, Xiaoming Sun, and Song Zuo. On the sensitivity complexity of bipartite graph properties. Theor. Comput. Sci., 468:83-91, 2013. Google Scholar
  19. Justin Gilmer, Michal Koucký, and Michael E. Saks. A new approach to the sensitivity conjecture. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 247-254, 2015. Google Scholar
  20. Parikshit Gopalan, Noam Nisan, Rocco A. Servedio, Kunal Talwar, and Avi Wigderson. Smooth boolean functions are easy: Efficient algorithms for low-sensitivity functions. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 59-70, 2016. Google Scholar
  21. Parikshit Gopalan, Rocco A. Servedio, and Avi Wigderson. Degree and sensitivity: Tails of two distributions. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 13:1-13:23, 2016. Google Scholar
  22. Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the sensitivity conjecture. Theory of Computing, Graduate Surveys, 4:1-27, 2011. Google Scholar
  23. Kun He, Qian Li, and Xiaoming Sun. A tighter relation between sensitivity and certificate complexity. CoRR, abs/1609.04342, 2016. Google Scholar
  24. Ilan Karpas. Lower bounds for sensitivity of graph properties. CoRR, abs/1609.05320, 2016. Google Scholar
  25. Raghav Kulkarni. Evasiveness through a circuit lens. In Innovations in Theoretical Computer Science, ITCS'13, Berkeley, CA, USA, January 9-12, 2013, pages 139-144, 2013. Google Scholar
  26. Raghav Kulkarni, Youming Qiao, and Xiaoming Sun. Any monotone property of 3-uniform hypergraphs is weakly evasive. Theor. Comput. Sci., 588:16-23, 2015. Google Scholar
  27. Raghav Kulkarni and Avishay Tal. On fractional block sensitivity. Chicago J. Theor. Comput. Sci., 2016, 2016. Google Scholar
  28. Chengyu Lin and Shengyu Zhang. Sensitivity conjecture and log-rank conjecture for functions with small alternating numbers. CoRR, abs/1602.06627, 2016. Google Scholar
  29. László Lovász and Neal E. Young. Lecture notes on evasiveness of graph properties. CoRR, cs.CC/0205031, 2002. Google Scholar
  30. Noam Nisan. Crew prams and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. Google Scholar
  31. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. In Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC'92, pages 462-467, 1992. Google Scholar
  32. Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theor. Comput. Sci., 3(3):371-384, 1976. Google Scholar
  33. Karthik C. S. and Sébastien Tavenas. On the sensitivity conjecture for disjunctive normal forms. CoRR, abs/1607.05189, 2016. Google Scholar
  34. Xiaoming Sun. Block sensitivity of weakly symmetric functions. Theor. Comput. Sci., 384(1):87-91, 2007. Google Scholar
  35. Xiaoming Sun. An improved lower bound on the sensitivity complexity of graph properties. Theoretical Computer Science, 412(29):3524-3529, 2011. Google Scholar
  36. Mario Szegedy. An o(n^0.4732) upper bound on the complexity of the GKS communication game. Electronic Colloquium on Computational Complexity (ECCC), 22:102, 2015. Google Scholar
  37. Avishay Tal. On the sensitivity conjecture. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 38:1-38:13, 2016. Google Scholar
  38. György Turán. The critical complexity of graph properties. Information Processing Letters, 18(3):151-153, 1984. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact