Irit Dinur
Abstract
The soundness error of a PCP verifier is the probability (usually denoted ε) that the verifier accepts an incorrect input. We are interested in the smallest possible values of ε for which the PCP theorem holds, and in particular whether the theorem holds when ε is an inverse polynomial function of the input length. We discuss the 'sliding scale conjecture' of [BGLR93, LY94] and related questions. We then sketch some of the existing approaches and constructions of PCPs with sub-constant soundness error.
- S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. Journal of the ACM, 45(3):501--555, 1998. Google Scholar
Digital Library
- S. Arora and M. Sudan. Improved low degree testing and its applications. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 485--495, El Paso, Texas, 4--6 May 1997. Google ScholarDigital Library
- S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70--122, 1998. Google ScholarDigital Library
- E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan. Robust PCPs of proximity, shorter PCPs and applications to coding. SICOMP Special Issue on Ran- domness and Computation, 36(4):889--974, 2006. Google ScholarDigital Library
- M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.Google Scholar
- M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs, and nonapproximability - towards tight results. SIAM Journal on Computing, 27(3):804--915, June 1998. Google ScholarDigital Library
- A. Bogdanov. Gap amplification fails below 1/2. Comment on ECCC TR05-046, can be found at http://eccc.uni-trier.de/eccc-reports/2005/TR05-046/commt01.pdf, 2005.Google Scholar
- I. Dinur, E. Fischer, G. Kindler, R. Raz, and S. Safra. PCP characterizations of NP: Towards a polynomially-small error-probability. In Proc. 31st ACM Symp. on Theory of Computing, 1999. Google ScholarDigital Library
- I. Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54(3), 2007. Google ScholarDigital Library
- I. Dinur and O. Reingold. Assignment testers: Towards a combinatorial proof of the PCP theorem. SICOMP Special Issue on Randomness and Computation, 36(4):975--1024, 2006. Google ScholarDigital Library
- U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. Journal of the ACM, 43(2):268--292, 1996. Google ScholarDigital Library
- U. Feige and J. Kilian. Impossibility results for recycling random bits in two-prover proof systems. In Proc. 27th ACM Symp. on Theory of Computing, pages 457--468, 1995. Google ScholarDigital Library
- J. Håstad. Some optimal inapproximability results. Journal of the ACM, 48:798--859, 2001. Google ScholarDigital Library
- C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM, 41(5):960--981, 1994. Google ScholarDigital Library
- D. Moshkovitz and R. Raz. Two query PCP with sub-constant error. In Proc. 49th IEEE Symp. on Foundations of Computer Science, 2008. To appear. Google ScholarDigital Library
- R. Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763--803, June 1998. Google ScholarDigital Library
- R. Raz and S. Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proc. 29th ACM Symp. on Theory of Computing, pages 475--484, 1997. Google ScholarDigital Library
- G. Tardos. Multi-prover encoding schemes and three-prover proof systems. Journal of Computer and System Sciences, 53(2):251--260, 1996. Google ScholarDigital Library
Index Terms
- PCPs with small soundness error
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