research-article

Infeasibility of instance compression and succinct PCPs for NP

Authors:

Rahul SanthanamAuthors Info & Claims

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

Pages 133 - 142

https://doi.org/10.1145/1374376.1374398

Published: 17 May 2008 Publication History

Abstract

The OR-SAT problem asks, given Boolean formulae Φ₁,...,Φ_m each of size at most n, whether at least one of the Φ_i's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. A number of parametric $\NP$ problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. An approach of Harnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (Buhrman-Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.

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Ben-David S(2024)Probabilistically Checkable Arguments for All NPAdvances in Cryptology – EUROCRYPT 202410.1007/978-3-031-58734-4_12(345-374)Online publication date: 1-May-2024
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Index Terms

Infeasibility of instance compression and succinct PCPs for NP
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image ACM Conferences

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

May 2008

712 pages

ISBN:9781605580470

DOI:10.1145/1374376

General Chair:
Richard Ladner
University of Washington
,
Program Chair:
Cynthia Dwork
Microsoft Research, Silicon Valley

Copyright © 2008 ACM.

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Published: 17 May 2008

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Ben-David S(2024)Probabilistically Checkable Arguments for All NPAdvances in Cryptology – EUROCRYPT 202410.1007/978-3-031-58734-4_12(345-374)Online publication date: 1-May-2024
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Luo W(2023)Polynomial Turing Compressions for Some Graph Problems Parameterized by Modular-WidthComputing and Combinatorics10.1007/978-3-031-49190-0_9(127-140)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-49190-0_9
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