Theory of Computing Systems

, Volume 49, Issue 2, pp 246–272 | Cite as

On the Complexity of Matroid Isomorphism Problem

Article

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. Using this, we are able to show that graphic matroid isomorphism testing for planar graphs can be done in deterministic polynomial time. 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 problems are 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.

Keywords

Computational complexity 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • B. V. Raghavendra Rao
    • 1
  • M. N. Jayalal Sarma
    • 2
  1. 1.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingChina

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