## Abstract

A fundamental goal of computational complexity (and foundations of cryptography) is to find a polynomial-time samplable distribution (e.g., the uniform distribution) and a language in NTIME(f(n)) for some polynomial function f, such that the language is hard on the average with respect to this distribution, given that NP is worst-case hard (i.e. NP ≠ P, or NP ⊈ BPP). Currently, no such result is known even if we relax the language to be in nondeterministic sub-exponential time. There has been a long line of research trying to explain our failure in proving such worst-case/average-case connections [FF93, Vio03, BT03, AGGM06]. The bottom line of this research is essentially that (under plausible assumptions) non-adaptive Turing reductions cannot prove such results. In this paper we revisit the problem. Our first observation is that the above mentioned negative arguments extend to a non-standard notion of average-case complexity, in which the distribution on the inputs with respect to which we measure the average-case complexity of the language, is only samplable in super-polynomial time. The significance of this result stems from the fact that in this non-standard setting, [GSTS05] did show a worst-case/average-case connection. In other words, their techniques give a way to bypass the impossibility arguments. By taking a closer look at the proof of [GSTS05], we discover that the worst-case/averagecase connection is proven by a reduction that "almost" falls under the category ruled out by the negative result. This gives rise to an intriguing new notion of (almost black-box) reductions. After extending the negative results to the non-standard average-case setting of [GSTS05], we ask whether their positive result can be extended to the standard setting, to prove some new worst-case/average-case connections. While we can not do that unconditionally, we are able to show that under a mild derandomization assumption, the worst-case hardness of NP implies the average-case hardness of NTIME(f(n)) (under the uniform distribution) where f is computable in quasi-polynomial time.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings |

Publisher | Springer Verlag |

Pages | 569-583 |

Number of pages | 15 |

ISBN (Print) | 9783540742074 |

DOIs | |

State | Published - 2007 |

Event | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 - Princeton, NJ, United States Duration: 20 Aug 2007 → 22 Aug 2007 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4627 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 |
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Country/Territory | United States |

City | Princeton, NJ |

Period | 20/08/07 → 22/08/07 |