eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-12-01
54:1
54:14
10.4230/LIPIcs.MFCS.2017.54
article
New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems
Allender, Eric
Hirahara, Shuichi
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol083-mfcs2017/LIPIcs.MFCS.2017.54/LIPIcs.MFCS.2017.54.pdf
computational complexity
Kolmogorov complexity
circuit size