computational complexity

, Volume 28, Issue 1, pp 113–144 | Cite as

Query-to-Communication Lifting for PNP

  • Mika Göös
  • Pritish Kamath
  • Toniann Pitassi
  • Thomas WatsonEmail author


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).


Query Communication Lifting PNP 

Subject classification



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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).


  1. Scott Aaronson, Greg Kuperberg & Christopher Granade (2017). Complexity Zoo. Online.
  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. IEEEGoogle Scholar
  3. Beigel, Richard: Perceptrons, PP, and the Polynomial Hierarchy. Computational complexity 4(4), 339–349 (1994)MathSciNetCrossRefGoogle Scholar
  4. Beigel, Richard, Reingold, Nick, Spielman, Daniel: PP Is Closed under Intersection. Journal of Computer and System Sciences 50(2), 191–202 (1995)MathSciNetCrossRefGoogle Scholar
  5. Andreas Blass & Yuri Gurevich: On the Unique Satisfiability Problem. Information and Control 55(1–3), 80–88 (1982)MathSciNetGoogle Scholar
  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. IEEEGoogle Scholar
  7. Harry Buhrman & Ronald de Wolf: Complexity Measures and Decision Tree Complexity: A Survey. Theoretical Computer Science 288(1), 21–43 (2002)MathSciNetCrossRefGoogle Scholar
  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 appearGoogle Scholar
  9. Siu On Chan, James Lee, Prasad Raghavendra & David Steurer (2016). Approximate Constraint Satisfaction Requires Large LP Relaxations. Journal of the ACM 63(4), 34:1–34:22Google Scholar
  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).
  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. IEEEGoogle Scholar
  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 DagstuhlGoogle Scholar
  13. Göös, Mika, Lovett, Shachar, Meka, Raghu, Watson, Thomas, Zuckerman, David: Rectangles Are Nonnegative Juntas. SIAM Journal on Computing 45(5), 1835–1869 (2016)MathSciNetCrossRefGoogle Scholar
  14. Göös, Mika, Pitassi, Toniann, Watson, Thomas: Deterministic Communication vs. Partition Number, SIAM Journal on Computing To appear (2018a)Google Scholar
  15. Göös, Mika, Pitassi, Toniann, Watson, Thomas: The Landscape of Communication Complexity Classes. Computational Complexity 27(2), 245–304 (2018b)MathSciNetCrossRefGoogle Scholar
  16. Johan Håstad, Stasys Jukna & Pavel Pudlák (1995). Top-Down Lower Bounds for Depth-Three Circuits. Computational Complexity 5(2), 99–112. ISSN 1420-8954Google Scholar
  17. Hatami, Hamed, Hosseini, Kaave, Lovett, Shachar: Structure of Protocols for XOR Functions. SIAM Journal on Computing 47(1), 208–217 (2018)MathSciNetCrossRefGoogle Scholar
  18. Russell Impagliazzo & Ryan Williams (2010). Communication Complexity with Synchronized Clocks. In Proceedings of the 25th Conference on Computational Complexity (CCC), 259–269. IEEEGoogle Scholar
  19. Stasys Jukna (2012). Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. SpringerGoogle Scholar
  20. Karchmer, Mauricio, Kushilevitz, Eyal, Nisan, Noam: Fractional Covers and Communication Complexity. SIAM Journal on Discrete Mathematics 8(1), 76–92 (1995)MathSciNetCrossRefGoogle Scholar
  21. Ko, Ker-I: Separating and Collapsing Results on the Relativized Probabilistic Polynomial-Time Hierarchy. Journal of the ACM 37(2), 415–438 (1990)MathSciNetCrossRefGoogle Scholar
  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. ACMGoogle Scholar
  23. Eyal Kushilevitz & Noam Nisan (1997). Communication Complexity. Cambridge University PressGoogle Scholar
  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. ACMGoogle Scholar
  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. IEEEGoogle Scholar
  26. Ramamohan Paturi & Janos Simon: Probabilistic Communication Complexity. Journal of Computer and System Sciences 33(1), 106–123 (1986)MathSciNetCrossRefGoogle Scholar
  27. Anup Rao & Amir Yehudayoff (2017). Communication Complexity. In preparationGoogle Scholar
  28. Ran Raz & Pierre McKenzie: Separation of the Monotone NC Hierarchy. Combinatorica 19(3), 403–435 (1999)MathSciNetCrossRefGoogle Scholar
  29. Alexander Razborov & Alexander Sherstov: The Sign-Rank of AC\(^0\). SIAM Journal on Computing 39(5), 1833–1855 (2010)MathSciNetCrossRefGoogle Scholar
  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. IEEEGoogle Scholar
  31. Rivest, Ronald: Learning Decision Lists. Machine Learning 2(3), 229–246 (1987)Google Scholar
  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. IEEEGoogle Scholar
  33. Santha, Miklos: Relativized Arthur-Merlin versus Merlin-Arthur Games. Information and Computation 80(1), 44–49 (1989)MathSciNetCrossRefGoogle Scholar
  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.
  35. Sherstov, Alexander: The Pattern Matrix Method. SIAM Journal on Computing 40(6), 1969–2000 (2011)MathSciNetCrossRefGoogle Scholar
  36. Yaoyun Shi & Yufan Zhu: Quantum Communication Complexity of Block-Composed Functions. Quantum Information and Computation 9(5–6), 444–460 (2009)MathSciNetGoogle Scholar
  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 DagstuhlGoogle Scholar
  38. Vadhan, Salil: Pseudorandomness. Foundations and Trends in Theoretical Computer Science 7(1–3), 1–336 (2012)MathSciNetCrossRefGoogle Scholar
  39. Nikolai Vereshchagin (1999). Relativizability in Complexity Theory. In Provability, Complexity, Grammars, volume 192 of AMS Translations, Series 2, 87–172. American Mathematical SocietyGoogle Scholar
  40. Ryan Williams (2001). Brute Force Search and Oracle-Based Computation. Technical report, Cornell University.
  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).
  42. Yannakakis, Mihalis: Expressing Combinatorial Optimization Problems by Linear Programs. Journal of Computer and System Sciences 43(3), 441–466 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018
corrected publication 2019

Authors and Affiliations

  • Mika Göös
    • 1
  • Pritish Kamath
    • 2
  • Toniann Pitassi
    • 3
  • Thomas Watson
    • 4
    Email author
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.University of TorontoTorontoCanada
  4. 4.University of MemphisMemphisUSA

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