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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

Funding

  • Girish, Uma: Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award, by the National Science Foundation grants No. CCF-1714779, CCF-2007462 and by the IBM Phd Fellowship.
  • Mittal, Kunal: Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
  • Raz, Ran: Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
  • Zhan, Wei: Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.

Acknowledgements

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

Cite AsGet BibTex

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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