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Query-to-Communication Lifting for P^NP

Authors Mika Göös, Pritish Kamath, Toniann Pitassi, Thomas Watson

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Mika Göös
Pritish Kamath
Toniann Pitassi
Thomas Watson

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Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for P^NP. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.CCC.2017.12

Abstract

We prove that the P^NP-type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P^NP-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 P^NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

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Keywords
  • Communication Complexity
  • Query Complexity
  • Lifting Theorem
  • P^NP

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References

  1. Scott Aaronson, Greg Kuperberg, and Christopher Granade. Complexity zoo. Online, 2017. URL: https://complexityzoo.uwaterloo.ca.
  2. László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. URL: http://dx.doi.org/10.1109/SFCS.1986.15.
  3. Richard Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational complexity, 4(4):339-349, 1994. URL: http://dx.doi.org/10.1007/BF01263422.
  4. Richard Beigel, Nick Reingold, and Daniel Spielman. PP is closed under intersection. Journal of Computer and System Sciences, 50(2):191-202, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1017.
  5. Andreas Blass and Yuri Gurevich. On the unique satisfiability problem. Information and Control, 55(1-3):80-88, 1982. URL: http://dx.doi.org/10.1016/S0019-9958(82)90439-9.
  6. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00144-X.
  7. Harry Buhrman, Nikolai Vereshchagin, and Ronald de Wolf. On computation and communication with small bias. In Proceedings of the 22nd Conference on Computational Complexity (CCC), pages 24-32. IEEE, 2007. URL: http://dx.doi.org/10.1109/CCC.2007.18.
  8. Mark Bun and Justin Thaler. Approximate degree and the complexity of depth three circuits. Technical Report TR16-121, Electronic Colloquium on Computational Complexity (ECCC), 2016. URL: https://eccc.weizmann.ac.il/report/2016/121/.
  9. Siu On Chan, James Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. Journal of the ACM, 63(4):34:1-34:22, 2016. URL: http://dx.doi.org/10.1145/2811255.
  10. Susanna de Rezende, Jakob Nordström, and Marc Vinyals. 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), pages 295-304. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.40.
  11. Mika Göös. Lower bounds for clique vs. independent set. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), pages 1066-1076. IEEE, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.69.
  12. Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for P^NP. Electronic Colloquium on Computational Complexity (ECCC), 24:24, 2017. URL: https://eccc.weizmann.ac.il/report/2017/024.
  13. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016. URL: http://dx.doi.org/10.1137/15M103145X.
  14. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), pages 1077-1088. IEEE, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
  15. Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), volume 55 of LIPIcs, pages 86:1-86:15. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86.
  16. Johan Håstad, Stasys Jukna, and Pavel Pudlák. Top-down lower bounds for depth-three circuits. Computational Complexity, 5(2):99-112, 1995. URL: http://dx.doi.org/10.1007/BF01268140.
  17. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 282-288. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.38.
  18. Russell Impagliazzo and Ryan Williams. Communication complexity with synchronized clocks. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 259-269. IEEE, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.32.
  19. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. Google Scholar
  20. Mauricio Karchmer, Eyal Kushilevitz, and Noam Nisan. Fractional covers and communication complexity. SIAM Journal on Discrete Mathematics, 8(1):76-92, 1995. URL: http://dx.doi.org/10.1137/S0895480192238482.
  21. Ker-I Ko. Separating and collapsing results on the relativized probabilistic polynomial-time hierarchy. Journal of the ACM, 37(2):415-438, 1990. URL: http://dx.doi.org/10.1145/77600.77623.
  22. Pravesh Kothari, Raghu Meka, and Prasad Raghavendra. 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). ACM, 2017. To appear. Google Scholar
  23. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  24. James Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the 47th Symposium on Theory of Computing (STOC), pages 567-576. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746599.
  25. Periklis Papakonstantinou, Dominik Scheder, and Hao Song. Overlays and limited memory communication. In Proceedings of the 29th Conference on Computational Complexity (CCC), pages 298-308. IEEE, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.37.
  26. Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. Journal of Computer and System Sciences, 33(1):106-123, 1986. URL: http://dx.doi.org/10.1016/0022-0000(86)90046-2.
  27. Anup Rao and Amir Yehudayoff. Communication Complexity. In preparation, 2017. Google Scholar
  28. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, 1999. URL: http://dx.doi.org/10.1007/s004930050062.
  29. Alexander Razborov and Alexander Sherstov. The sign-rank of AC⁰. SIAM Journal on Computing, 39(5):1833-1855, 2010. URL: http://dx.doi.org/10.1137/080744037.
  30. Ronald Rivest. Learning decision lists. Machine Learning, 2(3):229-246, 1987. URL: http://dx.doi.org/10.1007/BF00058680.
  31. Robert Robere, Toniann Pitassi, Benjamin Rossman, and Stephen Cook. Exponential lower bounds for monotone span programs. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 406-415. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.51.
  32. Miklos Santha. Relativized Arthur-Merlin versus Merlin-Arthur games. Information and Computation, 80(1):44-49, 1989. URL: http://dx.doi.org/10.1016/0890-5401(89)90022-9.
  33. Rocco Servedio, Li-Yang Tan, and Justin Thaler. Attribute-efficient learning and weight-degree tradeoffs for polynomial threshold functions. In Proceedings of the 25th Conference on Learning Theory (COLT), pages 14.1-14.19. JMLR, 2012. URL: http://www.jmlr.org/proceedings/papers/v23/servedio12/servedio12.pdf.
  34. Alexander Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969-2000, 2011. URL: http://dx.doi.org/10.1137/080733644.
  35. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum Information and Computation, 9(5-6):444-460, 2009. Google Scholar
  36. Justin Thaler. Lower bounds for the approximate degree of block-composed functions. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), volume 55 of LIPIcs, pages 17:1-17:15. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.17.
  37. Salil Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. URL: http://dx.doi.org/10.1561/0400000010.
  38. Nikolai Vereshchagin. Relativizability in complexity theory. In Provability, Complexity, Grammars, volume 192 of AMS Translations, Series 2, pages 87-172. American Mathematical Society, 1999. Google Scholar
  39. Ryan Williams. Brute force search and oracle-based computation. Technical report, Cornell University, 2001. URL: https://web.stanford.edu/~rrwill/bfsearch-rev.ps.
  40. Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441-466, 1991. URL: http://dx.doi.org/10.1016/0022-0000(91)90024-Y.
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