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On the Complexity of Matroid Isomorphism Problems

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Computer Science - Theory and Applications (CSR 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5675))

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Abstract

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in \(\Sigma_2^p\). In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be \(\Sigma_2^p\)-complete and is coNP-hard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism and matroid isomorphism are both polynomial time many-one equivalent to graph isomorphism.

We give a polynomial time Turing reduction from graphic matroid isomorphism problem to the graph isomorphism problem. We then give a polynomial time many-one reduction from bounded rank matroid isomorphism problem to graphic matroid isomorphism, thus showing that all the above problems are polynomial time equivalent.

Further, for linear and graphic matroids, we prove that the automorphism problem is polynomial time equivalent to the corresponding isomorphism problems. In addition, we give a polynomial time membership test algorithm for the automorphism group of a graphic matroid.

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Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, C.I.T. Campus, Chennai, 600 113, India

    Raghavendra Rao B.V.

  2. Institute for Theoretical Computer Science, Tsinghua University, Beijing, 100 084, China

    Jayalal Sarma M.N.

Authors
  1. Raghavendra Rao B.V.
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  2. Jayalal Sarma M.N.
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Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 4 Koptyug av, 630090, Novosibirsk, Russia

    Anna Frid

  2. Sobolev Institute of Mathematics SB RAS, Koptyug av. 4, 630090, Novosibirsk, Russia

    Andrey Morozov

  3. Max Planck Institute for Software Systems, Campus E1 4, 66123, Saarbrücken, Germany

    Andrey Rybalchenko

  4. Lehrstuhl für Theoretische Informatik, Institut für Informatik, Julius-Maximilians-Universität Würzburg, Am Hubland, 97974, Würzburg, Germany

    Klaus W. Wagner

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© 2009 Springer-Verlag Berlin Heidelberg

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Rao B.V., R., Sarma M.N., J. (2009). On the Complexity of Matroid Isomorphism Problems. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_27

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  • DOI: https://doi.org/10.1007/978-3-642-03351-3_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03350-6

  • Online ISBN: 978-3-642-03351-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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