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

Authors Eric Allender, Shuichi Hirahara

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Eric Allender
Shuichi Hirahara

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Eric Allender and Shuichi Hirahara. New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.MFCS.2017.54

Abstract

The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deals 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.

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 reductions.  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 large class of sets A.

* Hardness of MCSP^A implies hardness of MKTP^A for a wide class of
sets A. This is the first result directly relating the complexity of
MCSP^A and MKTP^A, for any A.

Subject Classification

Keywords
  • computational complexity
  • Kolmogorov complexity
  • circuit size

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