Irit Dinur
Abstract
The PCP theorem [Arora and Safra 1998; Arora et. al. 1998] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to Feige et al. [1996], has placed the PCP theorem at the heart of the area of inapproximability.
In this work, we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NP-hardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem.
Our approach is to consider the unsat value of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem.
The amplification lemma relies on a new notion of “graph powering” that can be applied to systems of binary constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander.
We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locally-testable codes whose length is linear up to a polylog factor, and whose correctness can be probabilistically verified by making a constant number of queries. Namely, we prove SAT ∈ PCP 1/2,1[log2(n⋅poly log n), O(1)].
Supplemental Material
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Correction to a reference in the original article (refereed material)
- 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.) Google Scholar
Digital Library
- 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. Google ScholarDigital Library
- Arora, S., and Safra, S. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1, 70--122. Google ScholarDigital Library
- Babai, L. 1985. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on Theory of Computing. ACM, New York, 421--429. Google ScholarDigital Library
- Babai, L., Fortnow, L., and Lund, C. 1991. Non-deterministic exponential time has two-prover interactive protocols. Computat. Complex. 1, 3--40.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google Scholar
- 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. Google ScholarDigital Library
- 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.Google Scholar
- 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. Google ScholarDigital Library
- Feige, U., Goldwasser, S., Lovász, L., Safra, S., and Szegedy, M. 1996. Approximating clique is almost NP-complete. J. ACM 43, 2, 268--292. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Fortnow, L., Rompel, J., and Sipser, M. 1994. On the power of multi-prover interactive protocols. Theoret. Comput. Sci. 134, 2, 545--557. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Goldreich, O. 1997. A sample of samplers a computational perspective on sampling. Electronic Colloquium on Computational Complexity TR97-020.Google Scholar
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Goldwasser, S., Micali, S., and Rackoff, C. 1989. The knowledge complexity of interactive proofs. SIAM J. Comput. 18, 186--208. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Håstad, J. 2001. Some optimal inapproximability results. J. ACM 48, 798--859. Google ScholarDigital Library
- Linial, N., and Wigderson, A. 2003. Expander graphs and their applications. Lecture notes of a course: http://www.math.ias.edu/boaz/ExpanderCourse/.Google Scholar
- Lund, C., Fortnow, L., Karloff, H., and Nisan, N. 1992. Algebraic methods for interactive proof systems. J. ACM 39, 4 (Oct.), 859--868. Google ScholarDigital Library
- O'Donnell, R., and Guruswami, V. 2005. Lecture notes from a course on the PCP theorem and hardness of approximation.Google Scholar
- Papadimitriou, C., and Yannakakis, M. 1991. Optimization, approximation and complexity classes. J. Comput. Syst. Sci. 43, 425--440.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Raz, R. 1998. A parallel repetition theorem. SIAM J. Comput. 27, 3 (June), 763--803. Google ScholarDigital Library
- Reingold, O. 2005. Undirected st-connectivity in log-space. In Proceedings of the 37th ACM Symposium on Theory of Computing. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Shamir, A. 1992. IP = PSPACE. J. ACM 39, 4 (Oct.), 869--877. (Preliminary version in 1990 FOCS, pp. 11--15.) Google ScholarDigital Library
- Sipser, M., and Spielman, D. A. 1996. Expander codes. IEEE Trans. Inform. Theory 42, 6, part 1, Codes and complexity, 1710--1722.Google ScholarCross Ref
Index Terms
- The PCP theorem by gap amplification
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Reviews
William A Fahle
The PCP theorem is important to approximation algorithms. For example, it allows one to determine that certain problems can have no polynomial-time approximation scheme (PTAS). This paper covers both the importance and origin of the PCP theorem, along with its implications. Dinur gives a very clear alternative proof of the PCP theorem, starting from the point of view of inapproximability. Each lemma is clearly explained, and logical steps of the proof are separated for clarity. Overall, the proof is given by iteration of graph expansion, which is defined in the paper. After the proof, further results about expander graphs, SAT, and its PCP class are provided. Important lemmas, such as the ability to clean up a constraint graph or the ability to amplify the inapproximability gap, are saved for later sections in the paper. This creates some confusion that could have been avoided by proceeding more chronologically with the proof. Likewise, additional results on short locally testable codes and assignment testers should have been covered in a separate work—this would have saved the paper from being 44 pages long. Online Computing Reviews Service
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