On the NP-Completeness of the Minimum Circuit Size Problem

Abstract

We study the Minimum Circuit Size Problem (MCSP): given the
truth-table of a Boolean function f and a number k, does there
exist a Boolean circuit of size at most k computing f? This is a
fundamental NP problem that is not known to be NP-complete. Previous
work has studied consequences of the NP-completeness of MCSP. We
extend this work and consider whether MCSP may be complete for NP
under more powerful reductions. We also show that NP-completeness of
MCSP allows for amplification of circuit complexity.
We show the following results.
- If MCSP is NP-complete via many-one reductions, the following circuit complexity amplification result holds: If NP cap co-NP requires 2^n^{Omega(1)-size circuits, then  E^NP requires 2^Omega(n)-size circuits.
 
- If MCSP is NP-complete under truth-table reductions, then
EXP neq NP cap SIZE(2^n^epsilon) for some epsilon> 0 and EXP neq ZPP. This result extends to polylog Turing reductions.