The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deal with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) within a factor of
n
1 −
o
(1)
is
indeed
NP-intermediate. To the best of our knowledge, these problems are the first natural NP-intermediate problems under the existence of an arbitrary one-way function. Our technique is quite general; we use it also to show that approximating the size of the largest clique in a graph within a factor of
n
1 −
o
(1)
is also NP-intermediate unless NP⊆ P/poly.
We also prove that MKTP is hard for the complexity class DET under non-uniform NC
0
reductions. This is surprising, since prior work on MCSP and MKTP had highlighted weaknesses of “local” reductions such as ≤
NC
0
m
. We exploit this local reduction to obtain several new consequences:
— MKTP is not in AC
0
[
p
].
— Circuit size lower bounds are equivalent to hardness of a relativized version MKTP
A
of MKTP under a class of uniform AC
0
reductions, for a significant class of sets
A
.
— Hardness of MCSP
A
implies hardness of MCSP
A
for a significant class of sets
A
. This is the first result directly relating the complexity of MCSP
A
and MCSP
A
, for any
A
.
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