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 .
Highlights
The Minimum Circuit Size Problem (MCSP) has attracted intense study over the years, because of its close connection with the natural proofs framework of Razborov and Rudich [36], and because it is a prominent candidate for NP-intermediate status
In [8] even stronger nonuniform consequences were shown to follow from the weaker assumption of hardness for TC0. (See Table 2.) In Corollary 4.13, we present a weaker uniform lower bound that follows from the weaker assumption that MCSP or MKTP is hard for TC0 under a more powerful notion of reducibility
We have advanced our understanding about MCSP and MKTP in the following two respects: (1) On one hand, under a very weak cryptographic assumption, the problem of approximating MCSP or MKTP is NP-intermediate under general types of reductions when the approximation factor is quite huge
Summary
The Minimum Circuit Size Problem (MCSP) has attracted intense study over the years, because of its close connection with the natural proofs framework of Razborov and Rudich [36], and because it is a prominent candidate for NP-intermediate status. Their proof relies on self-reducibility properties of the determinant, whereas our proof relies on the fact that Graph Isomorphism is hard for DET [42] Their results have the advantage that they apply to MCSP rather than merely to MKTP, but because their reduction is more complex (TC0, as contrasted with AC0), they do not obtain unconditional lower bounds, as in Corollary 4.4. A Clearer Picture of How Hardness ‘Evolves.” It is instructive to contrast the evolution of the class of problems reducible to MKTPA under different types of reductions, as A varies from very easy (A = ∅) to complex (A = QBF) For this thought experiment, we assume the very plausible hypothesis that DSPACE(n) io-SIZE(2εn ). We present a similar argument showing that a n1−o(1) approximation for CLIQUE is NP-intermediate if NP P/poly
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