Properties of NP‐Complete Sets
Christian Glaßer, A. Pavan, Alan L. Selman, Samik Sengupta · SIAM Journal on Computing · 2006
We study several properties of sets that are complete for NP. We prove that if L is an NP‐complete set and S ot\supseteq L is a p‐selective sparse set, then $L - S$ is $\leq^{p}_{m}$‐hard for NP. We demonstrate the existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$ such that for every $L \in \mbox{NP} - \mbox{P}$, L - S is not $\leq^p_m$‐hard for NP. Moreover, we prove for every $L \in \mbox{NP} - \mbox{P}$ that there exists a sparse $S \in $ EXP such that L - S is not $\leq^p_m$‐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$(2^{n^{\epsilon}})$‐immune. Also, using these hypotheses we show that no NP‐complete set is quasi‐polynomial‐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 the existence of a UP‐machine M that accepts $0^*$ such that for some $0 < \epsilon < 1$, no $2^{n^{\epsilon}}$ time‐bounded machine can correctly compute infinitely many accepting computations of M. We show that if $\UP \cap \co\UP$ contains DTIME$(2^{n^{\epsilon}})$‐bi‐immune sets, then this hypothesis is true.