Properties of NP-Complete Sets

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 resolve 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 ...

• Type: Conference
• Year: 2004
• Where: COCO
• Authors: Christian Glaßer, Aduri Pavan, Alan L. Selman, Samik Sengupta
• Keywords: COCO 2004, Np-complete Set, Secure One-way Permutations, Sparse Set