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Smooth and Strong PCPs

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A Correction to this article was published on 10 June 2021

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Abstract

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:

  • \(\circ \quad\)A PCP is strong if it rejects an alleged proof of a correct claimwith probability proportional to its distance from some correctproof of that claim.

  • \(\circ \quad\)A PCP is smooth if each location in a proof is queried with equalprobability.

We prove that all sets in \(\mathcal{NP}\) have PCPs that are both smooth andstrong, are of polynomial length and can be verified based on a constantnumber of queries. This is achieved by following the proof of thePCP theorem of Arora et al. (JACM 45(3):501–555, 1998), providing astronger analysis of the Hadamard and Reed–Muller based PCPs anda refined PCP composition theorem. In fact, we show that any set in \(\mathcal{NP}\) has a smooth strong canonical PCP of Proximity (PCPP), meaningthat there is an efficiently computable bijection of \(\mathcal{NP}\) witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth.

Our result implies the hardness of approximating the satisfiability of “stable” 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.

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References

  • Josh Alman & Lijie Chen (2019). Efficient Construction of Rigid Matrices Using an NP Oracle. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, MD, USA, November 9-12, 2019, David Zuckerman, editor, 1034–1055. IEEE Computer Society

  • Arora, Sanjeev, Lund, Carsten, Motwani, Rajeev, Sudan, Madhu, Szegedy, Mario: Proof Verification and the Hardness of Approximation Problems. J. ACM 45(3), 501–555 (1998)

    Article  MathSciNet  Google Scholar 

  • Sanjeev Arora & Shmuel Safra: Probabilistic Checking of Proofs: A New Characterization of NP. J. ACM 45(1), 70–122 (1998)

    Article  MathSciNet  Google Scholar 

  • Pranjal Awasthi, Avrim Blum & Or Sheffet (2010). Stability Yields a PTAS for k-Median and k-Means Clustering. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, 309–318. IEEE Computer Society. ISBN 978-0-7695-4244-7

  • Awasthi, Pranjal, Blum, Avrim, Sheffet, Or: Center-based clustering under perturbation stability. Inf. Process. Lett. 112(1–2), 49–54 (2012)

    Article  MathSciNet  Google Scholar 

  • Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy & Ali Kemal Sinop (2015). The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, Lars Arge & János Pach, editors, volume 34 of LIPIcs, 754–767. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. ISBN 978-3-939897-83-5

  • Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan & Salil P. Vadhan (2005). Short PCPs Verifiable in Polylogarithmic Time. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, 120–134. IEEE Computer Society. ISBN 0-7695-2364-1. URL https://doi.org/10.1109/CCC.2005.27

  • Ben-Sasson, Eli, Goldreich, Oded, Harsha, Prahladh, Sudan, Madhu, Vadhan, Salil P.: Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding. SIAM J. Comput. 36(4), 889–974 (2006)

    Article  MathSciNet  Google Scholar 

  • Eli Ben-Sasson & Madhu Sudan: Short PCPs with Polylog Query Complexity. SIAM J. Comput. 38(2), 551–607 (2008)

    Article  MathSciNet  Google Scholar 

  • Amey Bhangale, Prahladh Harsha, Orr Paradise & Avishay Tal (2020). Rigid Matrices From Rectangular PCPs. Electronic Colloquium on Computational Complexity (ECCC) 27, 75. URL https://eccc.weizmann.ac.il/report/2020/075

  • Yonatan Bilu & Nathan Linial: Lifts, Discrepancy and Nearly Optimal Spectral Gap*. Combinatorica 26(5), 495–519 (2006)

    Article  MathSciNet  Google Scholar 

  • Yonatan Bilu & Nathan Linial: Are Stable Instances Easy? Combinatorics, Probability & Computing 21(5), 643–660 (2012)

    Article  MathSciNet  Google Scholar 

  • Blum, Manuel, Luby, Michael, Rubinfeld, Ronitt: Self-Testing/Correcting with Applications to Numerical Problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)

    Article  MathSciNet  Google Scholar 

  • Shuchi Chawla (editor) (2020). Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020. SIAM. ISBN 978-1-61197-599-4. URL https://doi.org/10.1137/1.9781611975994

  • Alessandro Chiesa, Tom Gur & Igor Shinkar (2020). Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity. In DBLP:confspssodasps2020, 1395–1411. URL https://doi.org/10.1137/1.9781611975994.84

  • Dinur, Irit: The PCP theorem by gap amplification. J. ACM 54(3), 12 (2007)

    Article  MathSciNet  Google Scholar 

  • Irit Dinur, Eldar Fischer, Guy Kindler, Ran Raz & Shmuel Safra (2011). PCP Characterizations of NP: Toward a Polynomially-Small Error-Probability. Comput. Complex. 20(3), 413–504. URL https://doi.org/10.1007/s00037-011-0014-4

  • Irit Dinur, Oded Goldreich & Tom Gur (2019). Every Set in P Is Strongly Testable Under a Suitable Encoding. In 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, Avrim Blum, editor, volume 124 of LIPIcs, 30:1–30:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-095-8

  • Irit Dinur & Prahladh Harsha (2013). Composition of Low-Error 2-Query PCPs Using Decodable PCPs. SIAM J. Comput. 42(6), 2452–2486. URL https://doi.org/10.1137/100788161

  • Irit Dinur & Omer Reingold: Assignment Testers: Towards a Combinatorial Proof of the PCP Theorem. SIAM J. Comput. 36(4), 975–1024 (2006)

    Article  MathSciNet  Google Scholar 

  • Uriel Feige (1998). A Threshold of ln n for Approximating Set Cover. J. ACM 45(4), 634–652. URL https://doi.org/10.1145/285055.285059

  • Zachary Friggstad, Kamyar Khodamoradi & Mohammad R. Salavatipour (2019). Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-MEANS. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, Timothy M. Chan, editor, 2958–2972. SIAM. ISBN 978-1-61197-548-2

  • Anna Gál & Andrew Mills (2012). Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length. TOCT 3(2), 5:1–5:34

  • Oded Goldreich (2017). Introduction to Property Testing. Cambridge University Press. ISBN 978-1-107-19405-2

  • Oded Goldreich, Tom Gur & Ilan Komargodski (2019). Strong Locally Testable Codes with Relaxed Local Decoders. TOCT 11(3), 17:1–17:38

  • Goldreich, Oded, Karloff, Howard J., Schulman, Leonard J., Trevisan, Luca: Lower bounds for linear locally decodable codes and private information retrieval. Computational Complexity 15(3), 263–296 (2006)

    Article  MathSciNet  Google Scholar 

  • Oded Goldreich & Madhu Sudan: Locally testable codes and PCPs of almost-linear length. J. ACM 53(4), 558–655 (2006)

    Article  MathSciNet  Google Scholar 

  • Tom Gur & Oded Lachish (2020). On the Power of Relaxed Local Decoding Algorithms. In DBLP:confspssodasps2020, 1377–1394. URL https://doi.org/10.1137/1.9781611975994.83

  • Tom Gur, Govind Ramnarayan & Ron D. Rothblum (2018). Relaxed Locally Correctable Codes. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, Anna R. Karlin, editor, volume 94 of LIPIcs, 27:1–27:11. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-060-6

  • Gur, Tom, Rothblum, Ron D.: Non-interactive proofs of proximity. Computational Complexity 27(1), 99–207 (2018)

    Article  MathSciNet  Google Scholar 

  • Prahladh Harsha (2004). Robust PCPs of Proximity and Shorter PCPs. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA

  • Yuval Ishai & Mor Weiss (2014). Probabilistically Checkable Proofs of Proximity with Zero-Knowledge. In Theory of Cryptography - 11th Theory of Cryptography Conference, TCC 2014, San Diego, CA, USA, February 24-26, 2014. Proceedings, Yehuda Lindell, editor, volume 8349 of Lecture Notes in Computer Science, 121–145. Springer. ISBN 978-3-642-54241-1. URL https://doi.org/10.1007/978-3-642-54242-8_6

  • Jonathan Katz & Luca Trevisan (2000). On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, F. Frances Yao & Eugene M. Luks, editors, 80–86. ACM. ISBN 1-58113-184-4

  • Subhash Khot (2002). Hardness Results for Coloring 3 -Colorable 3 -Uniform Hypergraphs. In 43rd Symposium on Foundations of Computer Science (FOCS 2002), 16-19 November 2002, Vancouver, BC, Canada, Proceedings, 23–32. IEEE Computer Society. ISBN 0-7695-1822-2

  • Or Meir (2009). Combinatorial Construction of Locally Testable Codes. SIAM J. Comput. 39(2), 491–544. URL https://doi.org/10.1137/080729967

  • Thilo Mie (2009). Short PCPPs verifiable in polylogarithmic time with O(1) queries. Ann. Math. Artif. Intell. 56(3-4), 313–338. URL https://doi.org/10.1007/s10472-009-9169-y

  • Dana Moshkovitz & Ran Raz (2010). Two-query PCP with subconstant error. J. ACM 57(5), 29:1–29:29. URL https://doi.org/10.1145/1754399.1754402

  • Christos H. Papadimitriou & Mihalis Yannakakis (1991). Optimization, Approximation, and Complexity Classes. Journal of Computer and System Sciences 43(3), 425–440. ISSN 0022-0000

  • Orr Paradise (2019). Smooth and Strong PCPs. Electronic Colloquium on Computational Complexity (ECCC) 26, 23. URL https://eccc.weizmann.ac.il/report/2019/023

  • Orr Paradise (2020). Smooth and Strong PCPs. In 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, Thomas Vidick, editor, volume 151 of LIPIcs, 2:1–2:41. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. ISBN 978-3-95977-134-4. URL https://doi.org/10.4230/LIPIcs.ITCS.2020.2

  • Ran Raz & Shmuel Safra (1997). A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, Frank Thomson Leighton & Peter W. Shor, editors, 475–484. ACM. ISBN 0-89791-888-6. URL https://doi.org/10.1145/258533.258641

  • Luca Trevisan (2004). Inapproximability of Combinatorial Optimization Problems. Electronic Colloquium on Computational Complexity (ECCC) (065)

  • Emanuele Viola (2020). New lower bounds for probabilistic degree and AC0 with parity gates. Electronic Colloquium on Computational Complexity (ECCC) 27, 15. URL https://eccc.weizmann.ac.il/report/2020/015

  • Yekhanin, Sergey: Locally Decodable Codes. Foundations and Trends in Theoretical Computer Science 6(3), 139–255 (2012)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Electrical Engineering and Computer Sciences Department, University of California Berkeley, 396 Soda Hall #1776, Berkeley, CA, 94720-1776, USA

    Orr Paradise

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The original online version of this article was revised: On page 13, minor typo (missing reference) updated.

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Paradise, O. Smooth and Strong PCPs. comput. complex. 30, 1 (2021). https://doi.org/10.1007/s00037-020-00199-3

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  • DOI: https://doi.org/10.1007/s00037-020-00199-3

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