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Query-to-Communication Lifting for PNP

Abstract

We prove that the PNP-type query complexity (alternatively, decision list width) of any Boolean function f is quadratically related to the PNP-type communication complexity of a lifted version of f. As an application, we show that a certain “product” lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture PNP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

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  • 06 April 2019

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References

  1. Scott Aaronson, Greg Kuperberg & Christopher Granade (2017). Complexity Zoo. Online. https://complexityzoo.uwaterloo.ca

  2. László Babai, Peter Frankl & Janos Simon (1986). Complexity Classes in Communication Complexity Theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), 337–347. IEEE

  3. Beigel, Richard: Perceptrons, PP, and the Polynomial Hierarchy. Computational complexity 4(4), 339–349 (1994)

  4. Beigel, Richard, Reingold, Nick, Spielman, Daniel: PP Is Closed under Intersection. Journal of Computer and System Sciences 50(2), 191–202 (1995)

  5. Andreas Blass & Yuri Gurevich: On the Unique Satisfiability Problem. Information and Control 55(1–3), 80–88 (1982)

  6. Harry Buhrman, Nikolai Vereshchagin & Ronald de Wolf (2007). On Computation and Communication with Small Bias. In Proceedings of the 22nd Conference on Computational Complexity (CCC), 24–32. IEEE

  7. Harry Buhrman & Ronald de Wolf: Complexity Measures and Decision Tree Complexity: A Survey. Theoretical Computer Science 288(1), 21–43 (2002)

  8. Mark Bun & Justin Thaler (2018). Approximate Degree and the Complexity of Depth Three Circuits. In Proceedings of the 22nd International Conference on Randomization and Computation (RANDOM). To appear

  9. Siu On Chan, James Lee, Prasad Raghavendra & David Steurer (2016). Approximate Constraint Satisfaction Requires Large LP Relaxations. Journal of the ACM 63(4), 34:1–34:22

  10. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff & Sagnik Mukhopadhyay (2017). Composition and Simulation Theorems via Pseudo-random Properties. Technical Report TR17-014, Electronic Colloquium on Computational Complexity (ECCC). https://eccc.weizmann.ac.il/report/2017/014/

  11. Mika Göös (2015). Lower Bounds for Clique vs. Independent Set. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), 1066–1076. IEEE

  12. Mika Göös, Pritish Kamath, Toniann Pitassi & Thomas Watson (2017). Query-to-Communication Lifting for P NP. In Proceedings of the 32nd Computational Complexity Conference (CCC), 12:1–12:16. Schloss Dagstuhl

  13. Göös, Mika, Lovett, Shachar, Meka, Raghu, Watson, Thomas, Zuckerman, David: Rectangles Are Nonnegative Juntas. SIAM Journal on Computing 45(5), 1835–1869 (2016)

  14. Göös, Mika, Pitassi, Toniann, Watson, Thomas: Deterministic Communication vs. Partition Number, SIAM Journal on Computing To appear (2018a)

  15. Göös, Mika, Pitassi, Toniann, Watson, Thomas: The Landscape of Communication Complexity Classes. Computational Complexity 27(2), 245–304 (2018b)

  16. Johan Håstad, Stasys Jukna & Pavel Pudlák (1995). Top-Down Lower Bounds for Depth-Three Circuits. Computational Complexity 5(2), 99–112. ISSN 1420-8954

  17. Hatami, Hamed, Hosseini, Kaave, Lovett, Shachar: Structure of Protocols for XOR Functions. SIAM Journal on Computing 47(1), 208–217 (2018)

  18. Russell Impagliazzo & Ryan Williams (2010). Communication Complexity with Synchronized Clocks. In Proceedings of the 25th Conference on Computational Complexity (CCC), 259–269. IEEE

  19. Stasys Jukna (2012). Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer

  20. Karchmer, Mauricio, Kushilevitz, Eyal, Nisan, Noam: Fractional Covers and Communication Complexity. SIAM Journal on Discrete Mathematics 8(1), 76–92 (1995)

  21. Ko, Ker-I: Separating and Collapsing Results on the Relativized Probabilistic Polynomial-Time Hierarchy. Journal of the ACM 37(2), 415–438 (1990)

  22. Pravesh Kothari, Raghu Meka & Prasad Raghavendra (2017). Approximating Rectangles by Juntas and Weakly-Exponential Lower Bounds for LP Relaxations of CSPs. In Proceedings of the 49th Symposium on Theory of Computing (STOC), 590–603. ACM

  23. Eyal Kushilevitz & Noam Nisan (1997). Communication Complexity. Cambridge University Press

  24. James Lee, Prasad Raghavendra & David Steurer (2015). Lower Bounds on the Size of Semidefinite Programming Relaxations. In Proceedings of the 47th Symposium on Theory of Computing (STOC), 567–576. ACM

  25. Periklis Papakonstantinou, Dominik Scheder & Hao Song (2014). Overlays and Limited Memory Communication. In Proceedings of the 29th Conference on Computational Complexity (CCC), 298–308. IEEE

  26. Ramamohan Paturi & Janos Simon: Probabilistic Communication Complexity. Journal of Computer and System Sciences 33(1), 106–123 (1986)

  27. Anup Rao & Amir Yehudayoff (2017). Communication Complexity. In preparation

  28. Ran Raz & Pierre McKenzie: Separation of the Monotone NC Hierarchy. Combinatorica 19(3), 403–435 (1999)

  29. Alexander Razborov & Alexander Sherstov: The Sign-Rank of AC\(^0\). SIAM Journal on Computing 39(5), 1833–1855 (2010)

  30. Susanna de Rezende, Jakob Nordström & Marc Vinyals (2016). How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity). In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), 295–304. IEEE

  31. Rivest, Ronald: Learning Decision Lists. Machine Learning 2(3), 229–246 (1987)

  32. Robert Robere, Toniann Pitassi, Benjamin Rossman & Stephen Cook (2016). Exponential Lower Bounds for Monotone Span Programs. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), 406–415. IEEE

  33. Santha, Miklos: Relativized Arthur-Merlin versus Merlin-Arthur Games. Information and Computation 80(1), 44–49 (1989)

  34. Rocco Servedio, Li-Yang Tan & Justin Thaler (2012). Attribute-Efficient Learning and Weight-Degree Tradeoffs for Polynomial Threshold Functions. In Proceedings of the 25th Conference on Learning Theory (COLT), 14.1–14.19. JMLR. http://www.jmlr.org/proceedings/papers/v23/servedio12/servedio12.pdf

  35. Sherstov, Alexander: The Pattern Matrix Method. SIAM Journal on Computing 40(6), 1969–2000 (2011)

  36. Yaoyun Shi & Yufan Zhu: Quantum Communication Complexity of Block-Composed Functions. Quantum Information and Computation 9(5–6), 444–460 (2009)

  37. Justin Thaler (2016). Lower Bounds for the Approximate Degree of Block-Composed Functions. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), 17:1–17:15. Schloss Dagstuhl

  38. Vadhan, Salil: Pseudorandomness. Foundations and Trends in Theoretical Computer Science 7(1–3), 1–336 (2012)

  39. Nikolai Vereshchagin (1999). Relativizability in Complexity Theory. In Provability, Complexity, Grammars, volume 192 of AMS Translations, Series 2, 87–172. American Mathematical Society

  40. Ryan Williams (2001). Brute Force Search and Oracle-Based Computation. Technical report, Cornell University. https://web.stanford.edu/~rrwill/bfsearch-rev.ps

  41. Xiaodi Wu, Penghui Yao & Henry Yuen (2017). Raz–McKenzie Simulation with the Inner Product Gadget. Technical Report TR17-010, Electronic Colloquium on Computational Complexity (ECCC). https://eccc.weizmann.ac.il/report/2017/010/

  42. Yannakakis, Mihalis: Expressing Combinatorial Optimization Problems by Linear Programs. Journal of Computer and System Sciences 43(3), 441–466 (1991)

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Acknowledgements

We thank anonymous reviewers for comments, especially for a suggestion that led to a simplified proof of Claim 3.6. We thank Paul Balister, Shalev Ben-David, Béla Bollobás, Robin Kothari, Nirman Kumar, Santosh Kumar, Govind Ramnarayan, Madhu Sudan, Li-Yang Tan, and Justin Thaler for discussions and correspondence. T.W. was supported by NSF grant CCF-1657377. A preliminary version of this work was published as Göös et al. (2017).

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Correspondence to Thomas Watson.

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The original version of this article was revised: The inline images were not processed in the original version and updated here

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Göös, M., Kamath, P., Pitassi, T. et al. Query-to-Communication Lifting for PNP. comput. complex. 28, 113–144 (2019). https://doi.org/10.1007/s00037-018-0175-5

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Keywords

  • Query
  • Communication
  • Lifting
  • P NP

Subject classification

  • 68Q15