New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems

Abstract

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 .