Reduction from Non-Unique Games to Boolean Unique Games
We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-δ vs. 1-Cδ, for any C > 1, and sufficiently small δ > 0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known.
Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.
Unique Games Conjecture
hyperplane encoding
concentration of measure
low degree testing
Theory of computation~Problems, reductions and completeness
64:1-64:25
Regular Paper
https://arxiv.org/abs/2006.13073
https://eccc.weizmann.ac.il/report/2020/093/
We are thankful to Subhash Khot and Bo'az Klartag for discussions. We are especially grateful to Bo'az for suggesting to use Schur’s Lemma which is key to the proof of Theorem 8
Ronen
Eldan
Ronen Eldan
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Supported by a European Research Council Starting Grant (ERC StG) and by an Israel Science Foundation grant no. 715/16.
Dana
Moshkovitz
Dana Moshkovitz
Department of Computer Science, University of Texas at Austin, TX, USA
This material is based upon work supported by the National Science Foundation under grants number 1218547 and 1648712.
10.4230/LIPIcs.ITCS.2022.64
S. Arora, B. Barak, and D. Steurer. Subexponential algorithms for unique games and related problems. In Proc. 51st IEEE Symp. on Foundations of Computer Science, 2010.
S. Arora, S. A. Khot, A. Kolla, D. Steurer, M. Tulsiani, and N. Vishnoi. Unique games on expanding constraint graphs are easy: extended abstract. In Proc. 40th ACM Symp. on Theory of Computing, pages 21-28, 2008.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, 1998.
S. Arora and S. Safra. Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM, 45(1):70-122, 1998.
B. Barak, M. Hardt, I. Haviv, A. Rao, O. Regev, and D. Steurer. Rounding parallel repetitions of unique games. In Proc. 49th IEEE Symp. on Foundations of Computer Science, pages 374-383, 2008.
B. Barak, P. Kothari, and D. Steurer. Small-set expansion in shortcode graph and the 2-to-2 conjecture. In ITCS'19, 2019.
M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993.
A. Carbery and J. Wright. Distributional and l_q norm inequalities for polynomials over convex bodies in Rⁿ. Math. Res. Lett., 8(3):233-248, 2001.
S. O. Chan. Approximation resistance from pairwise independent subgroups. In Proc. 45th ACM Symp. on Theory of Computing, pages 447-456, 2013.
M. Charikar, K. Makarychev, and Y. Makarychev. Near-optimal algorithms for maximum constraint satisfaction problems. ACM Transactions on Algorithms, 5(3), 2009.
E. Chlamtac, K. Makarychev, and Y. Makarychev. How to play unique games using embeddings. In Proc. 47th IEEE Symp. on Foundations of Computer Science, pages 687-696, 2006.
I. Dinur, S. Khot, G. Kindler, D. Minzer, and S. Safra. On non-optimally expanding sets in grassmann graphs. In Proc. 50th ACM Symp. on Theory of Computing, 2018.
I. Dinur, S. Khot, G. Kindler, D. Minzer, and S. Safra. Towards a proof of the 2-to-1 games conjecture? In Proc. 50th ACM Symp. on Theory of Computing, pages 376-389, 2018.
R. Eldan. A two-sided estimate for the gaussian noise stability deficit. Invent. Math., 2014.
R. Eldan and B. Klartag. Pointwise estimates for marginals of convex bodies. J. Functional Analysis, 254(8):2275-2293, 2008.
M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115-1145, 1995.
A. Gupta and K. Talwar. Approximating unique games. In SODA, pages 99-106, 2006.
J. Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
J. Håstad, S. Huang, R. Manokaran, R. O'Donnell, and J. Wright. Improved NP-inapproximability for 2-variable linear equations. Theory of Computing, 13(19):1-51, 2017.
S. Khot. On the power of unique 2-prover 1-round games. In Proc. 34th ACM Symp. on Theory of Computing, pages 767-775, 2002.
S. Khot. On the unique games conjecture (invited survey). In IEEE Conference on Computational Complexity, pages 99-121, 2010.
S. Khot, G. Kindler, E. Mossel, and R. O'Donnell. Optimal inapproximability results for MAX-CUT and other two-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007.
S. Khot, D. Minzer, D. Moshkovitz, and S. Safra. Small set expansion in the johnson graph. Technical Report TR18-078, ECCC, 2018.
S. Khot, D. Minzer, and S. Safra. On independent sets, 2-to-2 games, and grassmann graphs. In Proc. 49th ACM Symp. on Theory of Computing, pages 576-589, 2017.
S. Khot, D. Minzer, and S. Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. In Proc. 59th IEEE Symp. on Foundations of Computer Science, 2018.
S. Khot and D. Moshkovitz. NP-hardness of approximately solving linear equations over reals. In Proc. 43rd ACM Symp. on Theory of Computing, pages 413-420, 2011.
S. Khot and D. Moshkovitz. Candidate hard unique game. In Proc. 48th ACM Symp. on Theory of Computing, pages 63-76, 2016.
S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-epsilon. Journal of Computer and System Sciences, 74(3):335-349, 2008.
B. Klartag and O. Regev. Quantum one-way communication can be exponentially stronger than classical communication. In Proc. 43rd ACM Symp. on Theory of Computing, pages 31-40, 2011.
A. Kolla, K. Makarychev, and Y. Makarychev. How to play unique games against a semi-random adversary: Study of semi-random models of unique games. In Proc. 52nd IEEE Symp. on Foundations of Computer Science, pages 443-452, 2011.
P. McCullagh. Tensor methods in statistics. Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1987.
D. Moshkovitz and R. Raz. Two query PCP with sub-constant error. Journal of the ACM, 57(5), 2010.
E. Mossel and J. Neeman. Robust dimension free isoperimetry in gaussian space. Annals of Probability, 43(3):971-991, 2015.
E. Mossel and J. Neeman. Robust optimality of gaussian noise stability. Journal of the European Math Society (JEMS), 17(2):433-482, 2015.
E. Mossel and J. Neeman. Noise stability and correlation with half spaces. Electron. J. Probab., 23(16), 2018.
R. O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
P. Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proc. 40th ACM Symp. on Theory of Computing, pages 245-254, 2008.
R. Raz. A parallel repetition theorem. In SIAM Journal on Computing, volume 27, pages 763-803, 1998.
R. Raz. A counterexample to strong parallel repetition. SIAM Journal on Computing, 40(3):771-777, 2011.
R. Rubinfeld and M. Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996.
L. Trevisan. Approximation algorithms for unique games. Theory of Computing, 4(1):111-128, 2008.
Ronen Eldan and Dana Moshkovitz
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