### Abstract

For every integer k > 0, and an arbitrarily small constant ε > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, non-adaptively queries 4k + k^{2} bits in the proof, accepts a correct proof with probability 1, i. e., it has perfect completeness, and accepts any supposed proof of a false statement with probability at most 2^{−} ^{k2} + ε. In particular, the verifier achieves optimal amortized query complexity of 1 + δ for arbitrarily small constant δ > 0. Such a characterization was already proved by Samorodnitsky and Trevisan (STOC 2000), but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier, we can decrease the number of query bits to 2k + k^{2}, equal to the number obtained by Samorodnitsky and Trevisan. Finally we extend some of the results to PCPs over non-Boolean alphabets.

Original language | English (US) |
---|---|

Pages (from-to) | 119-148 |

Number of pages | 30 |

Journal | Theory of Computing |

Volume | 1 |

DOIs | |

State | Published - Sep 28 2005 |

### Fingerprint

### Keywords

- Amortized query bits
- Amortized query bits
- Approximation algorithms
- Computational complexity
- Inapproximability
- PCP
- Perfect completeness
- Probabilistically checkable proofs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing*,

*1*, 119-148. https://doi.org/10.4086/toc.2005.v001a007

**Query efficient PCPs with perfect completeness.** / Håstad, Johan; Khot, Subhash.

Research output: Contribution to journal › Article

*Theory of Computing*, vol. 1, pp. 119-148. https://doi.org/10.4086/toc.2005.v001a007

}

TY - JOUR

T1 - Query efficient PCPs with perfect completeness

AU - Håstad, Johan

AU - Khot, Subhash

PY - 2005/9/28

Y1 - 2005/9/28

N2 - For every integer k > 0, and an arbitrarily small constant ε > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, non-adaptively queries 4k + k2 bits in the proof, accepts a correct proof with probability 1, i. e., it has perfect completeness, and accepts any supposed proof of a false statement with probability at most 2− k2 + ε. In particular, the verifier achieves optimal amortized query complexity of 1 + δ for arbitrarily small constant δ > 0. Such a characterization was already proved by Samorodnitsky and Trevisan (STOC 2000), but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier, we can decrease the number of query bits to 2k + k2, equal to the number obtained by Samorodnitsky and Trevisan. Finally we extend some of the results to PCPs over non-Boolean alphabets.

AB - For every integer k > 0, and an arbitrarily small constant ε > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, non-adaptively queries 4k + k2 bits in the proof, accepts a correct proof with probability 1, i. e., it has perfect completeness, and accepts any supposed proof of a false statement with probability at most 2− k2 + ε. In particular, the verifier achieves optimal amortized query complexity of 1 + δ for arbitrarily small constant δ > 0. Such a characterization was already proved by Samorodnitsky and Trevisan (STOC 2000), but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier, we can decrease the number of query bits to 2k + k2, equal to the number obtained by Samorodnitsky and Trevisan. Finally we extend some of the results to PCPs over non-Boolean alphabets.

KW - Amortized query bits

KW - Amortized query bits

KW - Approximation algorithms

KW - Computational complexity

KW - Inapproximability

KW - PCP

KW - Perfect completeness

KW - Probabilistically checkable proofs

UR - http://www.scopus.com/inward/record.url?scp=33748123082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748123082&partnerID=8YFLogxK

U2 - 10.4086/toc.2005.v001a007

DO - 10.4086/toc.2005.v001a007

M3 - Article

AN - SCOPUS:33748123082

VL - 1

SP - 119

EP - 148

JO - Theory of Computing

JF - Theory of Computing

SN - 1557-2862

ER -