The PCP theorem by gap amplification

Journal of the ACM

Volumn 54, Issue 3, 2007, Pages

  Dinur, Irit ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

Gap amplification; PCP

Indexed keywords

APPROXIMATION THEORY; COMBINATORIAL MATHEMATICS; COMPUTATIONAL COMPLEXITY; CONSTRAINT THEORY; FORMAL LANGUAGES; ITERATIVE METHODS; OPTIMIZATION;

COMBINATORIAL AMPLIFICATION TRANSFORMATION; CONSTRAINT SYSTEMS; GRAPH STRUCTURES;

THEOREM PROVING;

EID: 34250354807     PISSN: 00045411     EISSN: 1557735X     Source Type: Journal    
DOI: 10.1145/1236457.1236459     Document Type: Article

References (34)

1. ARORA, S. 1994. Probabilistic checking of proofs and the hardness of approximation problems. Ph.D. dissertation, U.C. Berkeley. (Available via anonymous ftp as Princeton TR94-476.)
2. ARORA, S., LUND, C., MOTWANI, R., SUDAN, M., AND SZEGEDY, M. 1998. Proof verification and intractability of approximation problems. J. ACM 45, 3, 501-555.
3. ARORA, S., AND SAFRA, S. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1, 70-122.
4. BASAI, L. 1985. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on Theory of Computing. ACM, New York, 421-429.
5. BASAI, L., FORTNOW, L., AND LUND, C. 1991. Non-deterministic exponential time has two-prover interactive protocols. Computat. Complex. 1, 3-40.
6. BEN-OR, M., GOLDWASSER, S., KILIAN, J., AND WIGDERSON, A. 1988. Multi prover interactive proofs: How to remove intractability assumptions. In Proceedings of the 20th ACM Symposium on Theory of Computing. ACM, New York, 113-131.
7. BEN-SASSON, E., GOLDREICH, O., HARSHA, P., SUDAN, M., AND VADHAN, S. 2004. Robust PCPs of proximity, shorter PCPs and applications to coding. In Proceedings of the 36th ACM Symposium on Theory of Computing. ACM, New York.
8. BEN-SASSON, E., AND SUDAN, M. 2004. Robust locally testable codes and products of codes. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science.
9. BEN-SASSON, E., SUDAN, M., VADHAN, S. P., AND WIGDERSON, A. 2003. Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In Proceedings of the 35th ACM Symposium on Theory of Computing. ACM, New York, 612-621.
10. BOGDANOV, A. 2005. 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.
11. DINUR, I., AND REINGOLD, O. 2004. Assignment testers: Towards combinatorial proofs of the PCP theorem. In Proceedings of the 45th Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, CA.
12. FEIGE, U., GOLDWASSER, S., LOVÁSZ, L., SAFRA, S., AND SZEGEDY, M. 1996. Approximating clique is almost NP-complete. J. ACM 43, 2, 268-292.
13. FEIGE, U., AND KILIAN, J. 1995. Impossibility results for recycling random bits in two-prover proof systems. In Proceedings of the 27th ACM Symposium on Theory of Computing. 457-468.
14. FORTNOW, L., ROMPEL, J., AND SIPSER, M. 1994. On the power of multi-prover interactive protocols. Theoret. Comput. Sci. 134, 2, 545-557.
15. FRIEDGUT, E., KALAI, G., AND NAOR, A. 2002. Boolean functions whose Fourier transform is concentrated on the first two levels. Adv. Appl. Math. 29, 427-437.
16. GOLDREICH, O. 1997. A sample of samplers a computational perspective on sampling. Electronic Colloquium on Computational Complexity TR97-020.
17. GOLDREICH, O., AND SAFRA, S. 1997. A combinatorial consistency lemma with application to proving the PCP theorem. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science. Lecture Notes in Computer Science, Springer-Verlag, New York.
18. GOLDREICH, O., AND SUDAN, M. 2002. Locally testable codes and PCPs of almost-linear length. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA, 13-22.
19. GOLDREICH, O., AND WIGDERSON, A. 1997. Tiny families of functions with random properties: A quality size trade off for hashing. J. Random Struct. Algorithms 11, 4, 315-343.
20. GOLDWASSER, S., MICALI, S., AND RACKOFF, C. 1989. The knowledge complexity of interactive proofs. SIAM J. Comput. 18, 186-208.
21. HARSHA, P., AND SUDAN, M. 2001. Small PCPs with low query complexity. In Proceedings of the 18th Annual Symposium of Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 2010. Springer-Verlag, New York, 327-338.
22. HÅSTAD, J. 2001. Some optimal inapproximability results. J. ACM 48, 798-859.
23. LINIAL, N., AND WIGDERSON, A. 2003. Expander graphs and their applications. Lecture notes of a course: http://www.math.ias. edu/boaz/ExpanderCourse/.
24. LUND, C., FORTNOW, L., KARLOFF, H., AND NISAN, N. 1992. Algebraic methods for interactive proof systems. J. ACM 39, 4 (Oct.), 859-868.
25. O'DONNELL, R., AND GURUSWAMI, V. 2005. Lecture notes from a course on the PCP theorem and hardness of approximation.
26. PAPADMITRIOU, C., AND YANNAKAKIS, M. 1991. Optimization, approximation and complexity classes. J. Comput. Syst. Sci. 43, 425-440.
27. POLISHCHUK, A., AND SPIELMAN, D. 1994. Nearly linear size holographic proofs. In Proceedings of the 26th ACM Symposium on Theory of Computing. ACM, New York, 194-203.
28. RADHAKRISHNAN, J. 2006. Gap amplification in PCPs using lazy random walks. In Proceedings of the 33rd International Colloqium on Automata, Languages, and Programming (Venice, Italy, July 10-14). Lecture Notes in Computer Science, vol. 4051. Springer-Verlag, New Yrok.
29. RAZ, R. 1998. A parallel repetition theorem. SIAM J. Comput. 27, 3 (June), 763-803.
30. REINGOLD, O. 2005. Undirected st-connectivity in log-space. In Proceedings of the 37th ACM Symposium on Theory of Computing.
31. REINGOLD, O., VADHAN, S., AND WIGDERSON, A. 2002. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. Ann. Math. 155, 1, 157-187.
32. SHAMIR, A. 1992. IP = PSPACE. J. ACM 39, 4 (Oct.), 869-877.
33. (Preliminary version in 1990 FOCS, pp. 11-15.)
34. SIPSER, M., AND SPIELMAN, D. A. 1996. Expander codes. IEEE Trans. Inform. Theory 42, 6, part 1, Codes and complexity, 1710-1722.