Towards a Proof of the $2$-to-$1$ Games Conjecture

by Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra

Theory of Computing, Volume 21(11), pp. 1-50, 2025

Bibliography with links to cited articles

[1]   Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2):317–331, 1997. [doi:10.1006/jcss.1997.1472]

[2]   Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy: Proof verification and the hardness of approximation problems. J. ACM, 45(3):501–555, 1998. [doi:10.1145/278298.278306]

[3]   Sanjeev Arora and Shmuel Safra: Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70–122, 1998. [doi:10.1145/273865.273901]

[4]   Sanjeev Arora and Madhu Sudan: Improved low-degree testing and its applications. Combinatorica, 23(3):365–426, 2003. [doi:10.1007/s00493-003-0025-0]

[5]   Per Austrin, Ryan O’Donnell, Li-Yang Tan, and John Wright: New NP-hardness results for 3-coloring and 2-to-1 label cover. ACM Trans. Comput. Theory, 6(1):2:1–20, 2014. [doi:10.1145/2537800]

[6]   Boaz Barak, Pravesh K. Kothari, and David Steurer: Small-set expansion in shortcode graph and the 2-to-2 conjecture. In Proc. 10th Innovations in Theoret. Comp. Sci. Conf. (ITCS’19), pp. 9:1–12. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2019. [doi:10.4230/LIPIcs.ITCS.2019.9]

[7]   Mihir Bellare, Oded Goldreich, and Madhu Sudan: Free bits, PCPs, and nonapproximability – Towards tight results. SIAM J. Comput., 27(3):804–915, 1998. [doi:10.1137/S0097539796302531]

[8]   Irit Dinur and Elazar Goldenberg: Locally testing direct products in the low error range. In Proc. 49th FOCS, pp. 613–622. IEEE Comp. Soc., 2008. [doi:10.1109/FOCS.2008.26]

[9]   Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra: Towards a proof of the 2-to-1 games conjecture? In Proc. 50th STOC, pp. 376–389. ACM Press, 2018. [doi:10.1145/3188745.3188804]

[10]   Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra: On non-optimally expanding sets in Grassmann graphs. Israel J. Math., 243(1):377–420, 2021. [doi:10.1007/s11856-021-2164-7]

[11]   Irit Dinur, Subhash Khot, Will Perkins, and Muli Safra: Hardness of finding independent sets in almost 3-colorable graphs. In Proc. 51st FOCS, pp. 212–221. IEEE Comp. Soc., 2010. [doi:10.1109/FOCS.2010.84]

[12]   Irit Dinur and Omer Reingold: Assignment testers: Towards combinatorial proofs of the PCP theorem. SIAM J. Comput., 36(4):975–1024, 2006. [doi:10.1137/S0097539705446962]

[13]   Irit Dinur and Samuel Safra: On the hardness of approximating minimum vertex cover. Ann. Math., 162(1):439–485, 2005. Accessible at JSTOR.

[14]   Irit Dinur and David Steurer: Analytical approach to parallel repetition. In Proc. 46th STOC, pp. 624–633. ACM Press, 2014. [doi:10.1145/2591796.2591884]

[15]   Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy: Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268–292, 1996. [doi:10.1145/226643.226652]

[16]   Venkatesan Guruswami, Johan Håstad, and Madhu Sudan: Hardness of approximate hypergraph coloring. SIAM J. Comput., 31(6):1663–1686, 2002. [doi:10.1137/S0097539700377165]

[17]   Johan Håstad: Clique is hard to approximate within n1-ϵ. Acta Math., 182:105–142, 1999. Preliminary version in FOCS’96. [doi:10.1007/BF02392825]

[18]   Johan Håstad: Some optimal inapproximability results. J. ACM, 48(4):798–859, 2001. [doi:10.1145/502090.502098]

[19]   Thomas Holenstein: Parallel repetition: Simplifications and the no-signaling case. Theory of Computing, 5(8):141–172, 2009. Preliminary version in STOC’07. [doi:10.4086/toc.2009.v005a008]

[20]   Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson: New direct-product testers and 2-query PCPs. SIAM J. Comput., 41(6):1722–1768, 2012. [doi:10.1137/09077299X]

[21]   Subhash Khot: On the power of unique 2-prover 1-round games. In Proc. 17th IEEE Conf. on Comput. Complexity (CCC’02), pp. 25–25. IEEE Comp. Soc., 2002. [doi:10.1109/CCC.2002.1004334]

[22]   Subhash Khot: Inapproximability of NP-complete problems, discrete Fourier analysis, and geometry. In Proceedings of the International Congress of Mathematicians 2010, pp. 2676–2697. World Scientific, 2010. Accessible at author’s website. [doi:10.1142/9789814324359_0163]

[23]   Subhash Khot: On the unique games conjecture. In Proc. 25th IEEE Conf. on Comput. Complexity (CCC’10), pp. 99–121. IEEE Comp. Soc., 2010. [doi:10.1109/CCC.2010.19]

[24]   Subhash Khot, Dor Minzer, Dana Moshkovitz, and Muli Safra: Small set expansion in the Johnson graph. Theory of Computing, 21(2):1–43, 2025. [doi:10.4086/toc.2025.v021a002, ECCC:TR18-078]

[25]   Subhash Khot, Dor Minzer, and Muli Safra: Pseudorandom sets in Grassmann graph have near-perfect expansion. Ann. Math., 198(1):1–92, 2023. [doi:10.4007/annals.2023.198.1.1]

[26]   Subhash Khot, Dor Minzer, and Muli Safra: On independent sets, 2-to-2 games, and Grassmann graphs. Theory of Computing, 21(10):1–56, 2025. Preliminary version in STOC’17. [doi:10.4086/toc.2025.v021a010]

[27]   Subhash Khot and Dana Moshkovitz: Candidate hard unique game. In Proc. 48th STOC, pp. 63–76. ACM Press, 2016. [doi:10.1145/2897518.2897531]

[28]   Subhash Khot and Ryan O’Donnell: SDP gaps and UGC-hardness for Max-Cut-Gain. Theory of Computing, 5(4):83–117, 2009. [doi:10.4086/toc.2009.v005a004]

[29]   Subhash Khot and Muli Safra: A two-prover one-round game with strong soundness. Theory of Computing, 9(28):863–887, 2013. [doi:10.4086/toc.2013.v009a028]

[30]   Ryan O’Donnell and John Wright: A new point of NP-hardness for unique games. In Proc. 44th STOC, pp. 289–306. ACM Press, 2012. [doi:10.1145/2213977.2214005]

[31]   Ran Raz: A parallel repetition theorem. SIAM J. Comput., 27(3):763–803, 1998. [doi:10.1137/S0097539795280895]

[32]   Ran Raz and Shmuel Safra: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proc. 29th STOC, pp. 475–484. ACM Press, 1997. [doi:10.1145/258533.258641]

[33]   Ronitt Rubinfeld and Madhu Sudan: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252–271, 1996. [doi:10.1137/S0097539793255151]

[34]   Luca Trevisan: On Khot’s Unique Games Conjecture. Bull. AMS, 49(1):91–111, 2012. AMS link.