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 /spl nsupe/ L is a p-selective sparse set, then L -S is ≤_m^p-hard for NP. We demonstrate existence of a sparse set S ∈ DTIME(2²ⁿ) such that for every L ∈ NP - P, L - S is not ≤_m^p-hard for NP. Moreover, we prove for every L ∈ NP - P, that there exists a sparse S ∈ EXP such that L - S is not ≤_m^p-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 long-standing 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(2^ne)-immune. Also, using these hypotheses we show that no NP-complete 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 < ε < 1, no 2^ne time-bounded machine can correctly compute infinitely many accepting computations of M, We show that if UP ∩ coUP contains DTIME(2^ne)-bi-immune sets, then this hypothesis is true.