Minimum Circuit Size, Graph Isomorphism, and Related Problems
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.
Reductions between NP-intermediate problems
Graph Isomorphism
Minimum Circuit Size Problem
time-bounded Kolmogorov complexity
20:1-20:20
Regular Paper
Eric
Allender
Eric Allender
Joshua A.
Grochow
Joshua A. Grochow
Dieter
van Melkebeek
Dieter van Melkebeek
Cristopher
Moore
Cristopher Moore
Andrew
Morgan
Andrew Morgan
10.4230/LIPIcs.ITCS.2018.20
Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM Journal on Computing, 35:1467-1493, 2006.
Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. In Mathematical Foundations of Computer Science 2014, pages 25-32, 2014.
Eric Allender, Joshua Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum circuit size, graph isomorphism, and related problems. Technical Report TR17-158, Electronic Colloquium on Computational Complexity, 2017.
Eric Allender and Shuichi Hirahara. New insights on the (non)-hardness of circuit minimization and related problems. In Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science, 2017. To appear.
Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences, 77:14-40, 2010.
László Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th annual ACM Symposium on Theory of Computing, pages 684-697, 2016.
László Babai, Paolo Codenotti, and Youming Qiao. Polynomial-time isomorphism test for groups with no abelian normal subgroups. In Automata, Languages, and Programming, pages 51-62, 2012.
László Babai. Local Expansion of Vertex-transitive Graphs and Random Generation in Finite Groups. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 164-174, 1991.
Marco Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Algorithms from natural lower bounds. In Proceedings of the 31st Computational Complexity Conference, pages 10:1-10:24, 2016.
Gudmund Skovbjerg Frandsen and Peter Bro Miltersen. Reviewing bounds on the circuit size of the hardest functions. Information Processing Letters, 95(2):354-357, 2005.
Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3):691-729, 1991.
Oded Goldreich, Amit Sahai, and Salil Vadhan. Can statistical zero knowledge be made non-interactive? or on the relationship of SZK and NISZK. In Advances in Cryptology - CRYPTO '99, pages 467-484, 1999.
Oded Goldreich and Salil Vadhan. Comparing entropies in statistical zero knowledge with applications to the structure of SZK. In Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pages 54-73, 1999.
Oded Goldreich and Salil Vadhan. On the complexity of computational problems regarding distributions. In Oded Goldreich, editor, Studies in Complexity and Cryptography - Miscellanea on the Interplay between Randomness and Computation, pages 13-29. Springer, 2011.
Joshua A. Grochow. Matrix Lie algebra isomorphism. In Proceedings of the 2012 IEEE Conference on Computational Complexity, pages 203-213, 2012.
Johan Håstad, Russell Impagliazzo, Leonid Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM Journal on Computing, 28:1364-1396, 1999.
Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of MCSP and its variants. In Proceedings of the 32nd Computational Complexity Conference, pages 7:1-7:20, 2017.
Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In Proceedings of the 31st Conference on Computational Complexity, pages 18:1-18:20, 2016.
Derek F. Holt, Bettina Eick, and Eamonn A. O'Brien. Handbook of Computational Group Theory. Discrete Mathematics and its Applications. Chapman &Hall/CRC, 2005.
Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proceedings of the 32nd ACM Symposium on Theory of Computing, pages 73-79, 2000.
Donald E. Knuth. The Art of Computer Programming, volume 3: Sorting and Searching. Addison-Wesley, 1973.
Johannes Köbler, Uwe Schöning, and Jacobo Torán. The Graph Isomorphism Problem: Its Structural Complexity. Birkhauser Verlag, Basel, Switzerland, Switzerland, 1993.
Cody D. Murray and R. Ryan Williams. On the (Non) NP-Hardness of Computing Circuit Complexity. Theory of Computing, 13:1-22, 2017.
Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994.
Igor C. Carboni Oliveira and Rahul Santhanam. Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness. In Proceedings of the 32nd Computational Complexity Conference, pages 18:1-18:49, 2017.
Erez Petrank and Ron M. Roth. Is code equivalence easy to decide? IEEE Transactions on Information Theory, 43:1602-1604, 1997.
Michael Rudow. Discrete logarithm and minimum circuit size. Information Processing Letters, 128:1-4, 2017.
Amit Sahai and Salil Vadhan. A complete problem for statistical zero knowledge. Journal of the ACM, 50(2):196-249, 2003.
Ákos Seress. Permutation group algorithms, volume 152 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2003.
Boris A. Trakhtenbrot. A survey of Russian approaches to perebor (brute-force searches) algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984.
Masaki Yamamoto. A tighter lower bound on the circuit size of the hardest Boolean functions. Technical Report TR11-086, Electronic Colloquium on Computational Complexity, 2011.
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