Abstract
We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard. If the group is Abelian, then the isomorphism problem is graph isomorphism hard. Moreover, if we impose the restriction that all equations are of bounded length, then we prove that the isomorphism problem for systems of equations over finite Abelian groups is graph isomorphism complete. Finally we prove that the problem of counting the number of isomorphisms of systems of equations is no harder than deciding whether there exist any isomorphisms at all.
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References
Agrawal, M., Thierauf, T.: The formula isomorphism problem. SIAM Journal on Computing 30(3), 990–1009 (2000)
Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: Equivalence and isomorphism for boolean constraint satisfaction. In: Conference for Computer Science Logic, pp. 412–426 (2002)
Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: The complexity of boolean constraint isomorphism. In: Proceedings of the 21st Symposium on Theoretical Aspects of Computer Science, pp. 164–175 (2004)
Booth, K.S., Colbourn, C.J.: Problems polynomially equivalent to graph isomorphism. Technical report, CS-77-04, Computer Science Dept., University of Waterloo (1979)
Goldmann, M., Russel, A.: The complexity of solving equations over finite groups. Information and Computation 178(1), 253–262 (2002)
Köbler, H., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Basel (1993)
Mathon, R.: A note on the graph isomorphism counting problem. Information Processing Letters 8(3), 131–132 (1979)
Moore, C., Tesson, P., Thérien, D.: Satisfiability of systems of equations over finite monoids. In: Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science, pp. 537–547 (2001)
Nordh, G., Jonsson, P.: The complexity of counting solutions to systems of equations over finite semigroups. In: Proceedings of the 10th International Computing and Combinatorics Conference (2004)
Tesson, P.: Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, School of Computer Science, McGill University, Montreal (2003)
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© 2004 Springer-Verlag Berlin Heidelberg
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Nordh, G. (2004). The Complexity of Equivalence and Isomorphism of Systems of Equations over Finite Groups. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28629-5_28
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DOI: https://doi.org/10.1007/978-3-540-28629-5_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22823-3
Online ISBN: 978-3-540-28629-5
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