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.
@InProceedings{hitchcock_et_al:LIPIcs.FSTTCS.2015.236, author = {Hitchcock, John M. and Pavan, A.}, title = {{On the NP-Completeness of the Minimum Circuit Size Problem}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {236--245}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.236}, URN = {urn:nbn:de:0030-drops-56613}, doi = {10.4230/LIPIcs.FSTTCS.2015.236}, annote = {Keywords: Minimum Circuit Size, NP-completeness, truth-table reductions, circuit complexity} }
Feedback for Dagstuhl Publishing