On the NP-Completeness of the Minimum Circuit Size Problem

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∩co-NP requires 2nΩ(1)-size circuits, then E requires 2-size circuits. • If MCSP is NP-complete under truth-table reductions, then EXP 6= NP ∩ SIZE(2 ) for some > 0 and EXP 6= ZPP. This result extends to polylog Turing reductions.