Abstract
In a projection PCP, also known as Label-Cover, the verifier makes two queries to the proof, and the answer to the first query determines at most one satisfying answer to the second query. Projection PCPs with low error probability are the basis of most NP-hardness of approximation results known today. In this essay we outline a construction of a projection PCP with low error and low blow-up. This yields sharp approximation thresholds and tight time lower bounds for approximation of a variety of problems, under an assumption on the time required for solving certain NP-hard problems exactly. The approach of the construction is algebraic, and it includes components such as low error, randomness-efficient low degree testing and composition of projection PCPs.
- S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501--555, 1998. 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
- S. Arora and M. Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365--426, 2003.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, 1998. Google ScholarDigital Library
- M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient probabilistically checkable proofs and applications to approximations. In Proc. 25th ACM Symp. on Theory of Computing, pages 294--304, 1993. 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. SIAM Journal on Computing, 36(4):889--974, 2006. Google ScholarDigital Library
- E. Ben-Sasson and M. Sudan. Short PCPs with polylog query complexity. SIAM Journal on Computing, 38:551--607, 2008. Google ScholarDigital Library
- I. Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54(3):12, 2007. Google ScholarDigital Library
- I. Dinur, E. Fischer, G. Kindler, R. Raz, and S. Safra. PCP characterizations of NP: Toward a polynomially-small error-probability. Computational Complexity, 20(3):413--504, 2011. Google ScholarDigital Library
- I. Dinur and P. Harsha. Composition of low-error 2-query PCPs using decodable PCPs. In Proc. 41st ACM Symp. on Theory of Computing, 2009.Google ScholarDigital Library
- U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. 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(4):798--859, 2001. Google ScholarDigital Library
- S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43:439--561, 2006.Google ScholarCross Ref
- S. Khot. Guest column: inapproximability results via long code based PCPs. SIGACT News, 36:25--42, June 2005. Google ScholarDigital Library
- C. Lund, L. Fortnow, H. J. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. In IEEE Symposium on Foundations of Computer Science, pages 2--10, 1990. Google ScholarDigital Library
- D. Moshkovitz. Lecture notes in probabilistically checkable proofs, MIT. http://people.csail.mit.edu/dmoshkov/courses/pcp-mit/index.html.Google Scholar
- D. Moshkovitz. An alternative proof of the Schwartz-Zippel lemma. Technical report, ECCC TR10-096, 2010.Google Scholar
- D. Moshkovitz. The projection games conjecture and the NP-hardness of ln n-approximating Set-Cover. Technical Report TR11-112, Electronic Colloquium on Computational Complexity (ECCC), 2011.Google Scholar
- D. Moshkovitz and R. Raz. Sub-constant error probabilistically checkable proof of almost-linear size. Technical Report TR07-026, Electronic Colloquium on Computational Complexity, 2007.Google Scholar
- D. Moshkovitz and R. Raz. Sub-constant error low degree test of almost-linear size. SIAM Journal on Computing, 38(1):140--180, 2008. Google ScholarDigital Library
- D. Moshkovitz and R. Raz. Two query PCP with sub-constant error. Journal of the ACM, 57(5), 2010. Google ScholarDigital Library
- C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes. Journal of Computer and System Sciences, 43:425--440, 1991.Google ScholarCross Ref
- M. Plotkin. Binary codes with specified minimum distance. IRE Transactions on Information Theory, 6:445--450, 1960.Google ScholarCross Ref
- R. Raz. A parallel repetition theorem. In SIAM Journal on Computing, volume 27, pages 763--803, 1998. Google ScholarDigital Library
- R. Raz and S. Safra. A sub-constant error-probability low-degree test and a sub-constant errorprobability PCP characterization of NP. In Proc. 29th ACM Symp. on Theory of Computing, pages 475--484, 1997. Google ScholarDigital Library
- R. Rubinfeld and M. Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252--271, 1996. Google ScholarDigital Library
- M. Sudan. Lecture notes in essential coding theory, MIT. http://people.csail.mit.edu/madhu/coding/course.htmlGoogle Scholar
- M. A. Tsfaman, S. G. Vlǎdut, and T. Zink. Modular curves, shimura curves, and codes better than the Varshamov-Gilbert bound. Math. Nachrichten, 109:21--28, 1982.Google ScholarCross Ref
Index Terms
- Guest column: algebraic construction of projection PCPs
Recommendations
Guest column: inapproximability results via Long Code based PCPs
Summer has come again. And what better way is there to spend a summer than to relax on a sandy beach, on a mountain top, or at a park's picnic tables, and... think theory! Summer is a particularly good time to attack the big questions whose openness ...
Guest Column: Algebraic Natural Proofs Ben Lee Volk
Algebraic Natural Proofs is a recent framework which formalizes the type of reasoning used for proving most lower bounds on algebraic computational models. This concept is similar to and inspired by the famous natural proofs notion of Razborov and ...
Column: Team Diagonalization
Ten years ago, Gla'er, Pavan, Selman, and Zhang [GPSZ08] proved that if P 6= NP, then all NP-complete sets can be simply split into two NP-complete sets. That advance might naturally make one wonder about a quite di erent potential consequence of NP-...
Comments