Summary: PROPERTIES OF NP-COMPLETE SETS
, A. PAVAN¡ , ALAN L. SELMAN¢ , AND SAMIK SENGUPTA£
Abstract. We study several properties of sets that are complete for ¤¦¥ . We prove that if § is an ¤¨¥ -complete
set and © § is a p-selective sparse set, then §© is ¨ -hard for ¤¦¥ . We demonstrate existence of a sparse
set ©!¦"¨#%$'&¨(0)214365 such that for every §7'¤¦¥89¥ , §9@© is not -hard for ¤¦¥ . Moreover, we prove for
every §@'¤¨¥!A¥ , that there exists a sparse ©B&DCE¥ such that §F!© is not ¨ -hard for ¤¨¥ . Hence, removing
sparse information in ¥ from a complete set leaves the set complete, while removing sparse information in &GCE¥
from a complete set may destroy its completeness. Previously, these properties were known only for exponential
time complexity classes.
We use hypotheses about pseudorandom generators and secure one-way permutations to derive consequences
for longstanding open questions about whether ¤¨¥ -complete sets are immune. For example, assuming that pseudo-
random generators and secure one-way permutations exist, it follows easily that ¤¨¥ -complete sets are not p-immune.
Assuming only that secure one-way permutations exist, we prove that no ¤¦¥ -complete set is E"H#%$'&¦(I)QPQRS5 -
immune. Also, using these hypotheses we show that no ¤¦¥ -complete set is quasipolynomial-close to ¥ .
We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing-complete sets for ¤¨¥ are
not closed under union. Our hypothesis asserts existence of a T¦¥ -machine U that accepts VXW such that for some
V`YabYdc , no ) P R time-bounded machine can correctly compute infinitely many accepting computations of U .
We show that if T¦¥FegfihpT¦¥ contains E"H#%$'&¨(0) P Rq5 -bi-immune sets, then this hypothesis is true.