Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs

Authors Uma Girish, Kunal Mittal, Ran Raz, Wei Zhan



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Author Details

Uma Girish
  • Princeton University, NJ, USA
Kunal Mittal
  • Princeton University, NJ, USA
Ran Raz
  • Princeton University, NJ, USA
Wei Zhan
  • Princeton University, NJ, USA

Acknowledgements

We thank Justin Holmgren for important conversations and collaboration in early stages of this work.

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Uma Girish, Kunal Mittal, Ran Raz, and Wei Zhan. Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.6

Abstract

We prove that for every 3-player (3-prover) game G with value less than one, whose query distribution has the support S = {(1,0,0), (0,1,0), (0,0,1)} of Hamming weight one vectors, the value of the n-fold parallel repetition G^{⊗n} decays polynomially fast to zero; that is, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every 3-player game G over binary questions and arbitrary answer lengths, with value less than 1, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}. Our proof technique is new and requires many new ideas. For example, we make use of the Level-k inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Parallel repetition
  • Multi-prover games
  • Fourier analysis

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References

  1. Noga Alon and Bo'az Klartag. Economical toric spines via Cheeger’s inequality. J. Topol. Anal., 1(2):101-111, 2009. Google Scholar
  2. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM J. Comput., 42(3):1327-1363, 2013. (also in STOC 2010). Google Scholar
  3. Boaz Barak, Anup Rao, Ran Raz, Ricky Rosen, and Ronen Shaltiel. Strong parallel repetition theorem for free projection games. In APPROX-RANDOM, pages 352-365, 2009. Google Scholar
  4. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCPs, and nonapproximability - towards tight results. SIAM J. Comput., 27(3):804-915, 1998. (also in FOCS 1995). Google Scholar
  5. Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In STOC, pages 113-131, 1988. Google Scholar
  6. Mark Braverman and Ankit Garg. Small value parallel repetition for general games. In STOC, pages 335-340, 2015. Google Scholar
  7. Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In FOCS, pages 746-755, 2013. Google Scholar
  8. Richard Cleve, Peter Høyer, Benjamin Toner, and John Watrous. Consequences and limits of nonlocal strategies. In CCC, pages 236-249, 2004. Google Scholar
  9. Irit Dinur, Prahladh Harsha, Rakesh Venkat, and Henry Yuen. Multiplayer parallel repetition for expanding games. In ITCS, volume 67 of LIPIcs, pages Art. No. 37, 16, 2017. Google Scholar
  10. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In STOC, pages 624-633, 2014. Google Scholar
  11. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. (also in STOC 1996). Google Scholar
  12. Uriel Feige, Guy Kindler, and Ryan O'Donnell. Understanding parallel repetition requires understanding foams. In CCC, pages 179-192, 2007. Google Scholar
  13. Uriel Feige and Oleg Verbitsky. Error reduction by parallel repetition - a negative result. Combinatorica, 22(4):461-478, 2002. Google Scholar
  14. Lance Fortnow, John Rompel, and Michael Sipser. On the power of multi-prover interactive protocols. Theoret. Comput. Sci., 134(2):545-557, 1994. Google Scholar
  15. H. Furstenberg and Y. Katznelson. A density version of the Hales-Jewett theorem. J. Anal. Math., 57:64-119, 1991. Google Scholar
  16. Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, and Wei Zhan. Parallel repetition for the GHZ game: A simpler proof. In APPROX-RANDOM, pages 62:1-62:19, 2021. Google Scholar
  17. Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, and Wei Zhan. Parallel repetition for all 3-player games over binary alphabet. In STOC, 2022. Google Scholar
  18. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. (also in STOC 1997). Google Scholar
  19. Jan Hazla, Thomas Holenstein, and Anup Rao. Forbidden subgraph bounds for parallel repetition and the density hales-jewett theorem. CoRR, abs/1604.05757, 2016. URL: http://arxiv.org/abs/1604.05757.
  20. Thomas Holenstein. Parallel repetition: simplifications and the no-signaling case. Theory Comput., 5:141-172, 2009. (also in STOC 2007). Google Scholar
  21. Justin Holmgren and Ran Raz. A parallel repetition theorem for the GHZ game. CoRR, abs/2008.05059, 2020. URL: http://arxiv.org/abs/2008.05059.
  22. Justin Holmgren and Lisa Yang. The parallel repetition of non-signaling games: counterexamples and dichotomy. In STOC, pages 185-192, 2019. Google Scholar
  23. Guy Kindler, Ryan O'Donnell, Anup Rao, and Avi Wigderson. Spherical cubes and rounding in high dimensions. In FOCS, pages 189-198, 2008. Google Scholar
  24. Kunal Mittal and Ran Raz. Block rigidity: strong multiplayer parallel repetition implies super-linear lower bounds for Turing machines. In ITCS, pages Art. No. 71, 15, 2021. Google Scholar
  25. Michael Mitzenmacher and Eli Upfal. Probability and computing. Cambridge University Press, Cambridge, 2005. Randomized algorithms and probabilistic analysis. Google Scholar
  26. Ryan O'Donnell. Analysis of Boolean functions. Cambridge University Press, New York, 2014. Google Scholar
  27. Itzhak Parnafes, Ran Raz, and Avi Wigderson. Direct product results and the GCD problem, in old and new communication models. In STOC, pages 363-372, 1997. Google Scholar
  28. D. H. J. Polymath. A new proof of the density Hales-Jewett theorem. Ann. of Math. (2), 175(3):1283-1327, 2012. Google Scholar
  29. Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM J. Comput., 40(6):1871-1891, 2011. (also in STOC 2008). Google Scholar
  30. Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. (also in STOC 1995). Google Scholar
  31. Ran Raz. Parallel repetition of two prover games. In CCC, pages 3-6, 2010. Google Scholar
  32. Ran Raz and Ricky Rosen. A strong parallel repetition theorem for projection games on expanders. In CCC, pages 247-257, 2012. Google Scholar
  33. Oleg Verbitsky. Towards the parallel repetition conjecture. Theoret. Comput. Sci., 157(2):277-282, 1996. Google Scholar
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