Reductions to Graph Isomorphism

AbstractWe show that several reducibility notions coincide when applied to the Graph Isomorphism (GI) problem. In particular we show that if a set is many-one logspace reducible to GI, then it is in fact many-one $\textsf{AC}^{0}$ reducible to GI. For the case of Turing reducibilities we show that for any k≥0 an $\textsf{NC}^{k+1}$ reduction to GI can be transformed into an $\textsf{AC}^{k}$ reduction to the same problem.

[1]  Jacobo Torán,et al.  Completeness results for graph isomorphism , 2003, J. Comput. Syst. Sci..

[2]  Christopher B. Wilson.  On the Decomposability of NC and AC , 1990, SIAM J. Comput..

[3]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[4]  Janos Simon,et al.  Space-Bounded Hierarchies and Probabilistic Computations , 1984, J. Comput. Syst. Sci..

[5]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[6]  Christopher B. Wilson.  Decomposing NC and AC , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[7]  Jacobo Torán,et al.  On the hardness of graph isomorphism , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[8]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[9]  Mitsunori Ogihara.  Equivalence of NC^k and AC^k-1 closures of NP and Other Classes , 1995, Inf. Comput..

[10]  Alan L. Selman,et al.  Promise Problems Complete for Complexity Classes , 1988, Inf. Comput..

[11]  Eric Allender,et al.  Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem , 1998, J. Comput. Syst. Sci..

[12]  Christoph M. Hoffmann,et al.  Group-Theoretic Algorithms and Graph Isomorphism , 1982, Lecture Notes in Computer Science.

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  José L. Balcázar,et al.  Adaptive logspace reducibility and parallel time , 2005, Mathematical systems theory.

[16]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

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