Graph Isomorphism and Circuit Size

It is well-known [KST93] that the complexity of the Graph Automorphism problem is characterized by a special case of Graph Isomorphism, where the input graphs satisfy the "promise" of being rigid (that is, having no nontrivial automorphisms). In this brief note, we observe that the reduction of Graph Automorphism to the Rigid Graph Ismorphism problem can be accomplished even using Grollman and Selman's notion of a "smart reduction".

[1]  G. Chaitin,et al.  Kolmogorov complexity , 2010 .

[2]  Christian Glaßer,et al.  Reductions between disjoint NP-pairs , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[3]  Eric Allender,et al.  The Minimum Oracle Circuit Size Problem , 2016, computational complexity.

[4]  Ker-I Ko,et al.  Some Observations on the Probabilistic Algorithms and NP-hard Problems , 1982, Inf. Process. Lett..

[5]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[6]  Eric Allender,et al.  Power from Random Strings , 2006, SIAM J. Comput..

[7]  Eric Allender,et al.  The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory , 2009, J. Comput. Syst. Sci..

[8]  Boris A. Trakhtenbrot,et al.  A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms , 1984, Annals of the History of Computing.

[9]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[10]  Christophe-André Irniger.  Graph matching: filtering databases of graphs using machine learning techniques , 2005 .

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

[12]  Eric Allender,et al.  Power from random strings , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[13]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

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

[15]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[16]  Christian Glaßer,et al.  A Protocol for Serializing Unique Strategies , 2004, MFCS.

[17]  Leonid A. Levin,et al.  A Pseudorandom Generator from any One-way Function , 1999, SIAM J. Comput..

[18]  Cody Murray,et al.  On the (Non) NP-Hardness of Computing Circuit Complexity , 2015, Theory Comput..