## Properties of NP-complete sets (2004)

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Venue: | In Proceedings of the 19th IEEE Conference on Computational Complexity |

Citations: | 10 - 6 self |

### BibTeX

@INPROCEEDINGS{Glaßer04propertiesof,

author = {Christian Glaßer and A. Pavan and Alan L. Selman and Samik Sengupta},

title = {Properties of NP-complete sets},

booktitle = {In Proceedings of the 19th IEEE Conference on Computational Complexity},

year = {2004},

pages = {184--197},

publisher = {IEEE Computer Society}

}

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### Abstract

We study several properties of sets that are complete for NP. We prove that if L is an NP-complete set and S � ⊇ L is a p-selective sparse set, then L − S is ≤p m-hard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p m-hard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p m-hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one-way permutations to derive consequences for longstanding open questions about whether NP-complete sets are immune. For example, assuming that pseudorandom generators and secure one-way permutations exist, it follows easily that NP-complete sets are not p-immune. Assuming only that secure one-way permutations exist, we prove that no NP-complete set is DTIME(2nɛ)-immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomial-close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing-complete sets for NP are not closed under union. Our hypothesis asserts existence of a UP-machine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ time-bounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)-bi-immune sets, then this hypothesis is true.