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 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.

[1]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[2]  Amihood Amir,et al.  Some connections between bounded query classes and non-uniform complexity , 2000, Inf. Comput..

[3]  John M. Hitchcock,et al.  Comparing Reductions to NP-Complete Sets , 2006, ICALP.

[4]  Juris Hartmanis,et al.  On Isomorphisms and Density of NP and Other Complete Sets , 1977, SIAM J. Comput..

[5]  Cody Murray,et al.  On the (Non) NP-Hardness of Computing Circuit Complexity , 2017, Theory Comput..

[6]  Amihood Amir,et al.  Some connections between bounded query classes and nonuniform complexity , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[7]  Uwe Schöning.  The power of counting , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[8]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[9]  Eric Allender,et al.  Zero knowledge and circuit minimization , 2014, Inf. Comput..

[10]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[11]  J. Davenport.  Editor , 1960 .

[12]  Eric Allender,et al.  The Minimum Oracle Circuit Size Problem , 2015, STACS.

[13]  Harry Buhrman,et al.  Non-Uniform Reductions , 2008, Theory of Computing Systems.

[14]  Rahul Santhanam,et al.  Making hard problems harder , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[15]  Timothy J. Long.  Strong Nondeterministic Polynomial-Time Reducibilities , 1982, Theor. Comput. Sci..

[16]  Jin-Yi Cai,et al.  Circuit minimization problem , 2000, STOC '00.

[17]  Manindra Agrawal,et al.  Pseudo-random generators and structure of complete degrees , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[18]  John M. Hitchcock,et al.  Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses , 2010, STACS.