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On the (Non) NP-Hardness of Computing Circuit Complexity

Authors Cody D. Murray, R. Ryan Williams

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Cody D. Murray
R. Ryan Williams

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Cody D. Murray and R. Ryan Williams. On the (Non) NP-Hardness of Computing Circuit Complexity. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 365-380, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


The Minimum Circuit Size Problem (MCSP) is: given the truth table of a Boolean function f and a size parameter k, is the circuit complexity of f at most k? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of its kind, MCSP is not known to be NP-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, MCSP in P would imply there are no pseudorandom functions. Although most NP-complete problems are complete under strong "local" reduction notions such as poly-logarithmic time projections, we show that MCSP is provably not NP-hard under O(n^(1/2-epsilon))-time projections, for every epsilon > 0. We prove that the NP-hardness of MCSP under (logtime-uniform) AC0 reductions would imply extremely strong lower bounds: NP \not\subset P/poly and E \not\subset i.o.-SIZE(2^(delta * n)) for some delta > 0 (hence P = BPP also follows). We show that even the NP-hardness of MCSP under general polynomial-time reductions would separate complexity classes: EXP != NP \cap P/poly, which implies EXP != ZPP. These results help explain why it has been so difficult to prove that MCSP is NP-hard. We also consider the nondeterministic generalization of MCSP: the Nondeterministic Minimum Circuit Size Problem (NMCSP), where one wishes to compute the nondeterministic circuit complexity of a given function. We prove that the Sigma_2 P-hardness of NMCSP, even under arbitrary polynomial-time reductions, would imply EXP \not\subset P/poly.
  • circuit lower bounds
  • Minimum Circuit Size Problem
  • NP-completeness
  • projections
  • Reductions


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  1. Manindra Agrawal, Eric Allender, Russell Impagliazzo, Toniann Pitassi, and Steven Rudich. Reducing the complexity of reductions. Computational Complexity, 10(2):117-138, 2001. Google Scholar
  2. Eric Allender. When worlds collide: Derandomization, lower bounds, and Kolmogorov complexity. In FSTTCS, volume 2245 of LNCS, pages 1-15. Springer, 2001. Google Scholar
  3. Eric Allender. Personal communication, 2014. Google Scholar
  4. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. Google Scholar
  5. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. In Mathematical Foundations of Computer Science (MFCS), Part II, pages 25-32, 2014. Google Scholar
  6. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael Saks. Minimizing disjunctive normal form formulas and AC⁰ circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. Google Scholar
  7. Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In STACS, pages 21-33, 2015. Google Scholar
  8. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. Google Scholar
  9. Harry Buhrman, Lance Fortnow, and Thomas Thierauf. Nonrelativizing separations. In Proceedings of 13th Annual IEEE Conf. Computational Complexity, pages 8-12, 1998. Google Scholar
  10. Sebastian Czort. The complexity of minimizing disjunctive normal form formulas. Master’s Thesis, University of Aarhus, 1999. Google Scholar
  11. Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. J. ACM, 38(3):691-729, 1991. Google Scholar
  12. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, pages 6-20, 1986. Google Scholar
  13. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. JCSS, 65(4):672-694, 2002. Google Scholar
  14. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, pages 220-229, 1997. Google Scholar
  15. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In STOC, pages 73-79, 2000. Google Scholar
  16. Christos H. Papadimitriou and Mihalis Yannakakis. A note on succinct representations of graphs. Information and Control, 71(3):181-185, 1986. Google Scholar
  17. Sven Skyum and Leslie G Valiant. A complexity theory based on boolean algebra. J. ACM, 32(2):484-502, 1985. Google Scholar
  18. N. V. Vinodchandran. Nondeterministic circuit minimization problem and derandomizing Arthur-Merlin games. International Journal of Foundations of Computer Science, 16(6):1297-1308, 2005. Google Scholar
  19. Ryan Williams. Natural proofs versus derandomization. In STOC, pages 21-30, 2013. Google Scholar
  20. Stanislav Žák. A Turing machine time hierarchy. Theoretical Computer Science, 26(3):327-333, 1983. Google Scholar
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