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 co NP-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.
Highlights
Isomorphism problems over various mathematical structures have been a source of intriguing problems in complexity theory
We show that coloured version of the linear matroid isomorphism and graphic matroid isomorphisms are as hard as the general version (Lemma 4.2, 4.1)
We studied the matroid isomorphism problem under various input representations and restriction on the rank of the matroid
Summary
Isomorphism problems over various mathematical structures have been a source of intriguing problems in complexity theory (see [AT05]). From the recent results on graph isomorphism problem for these classes of graphs [DLN08, TW08], it follows that graphic matroid isomorphism problem for 3-connected planar graphs L-complete In this context we study the general, linear and graphic matroid isomorphism problems. As an immediate application of this, we show that the automorphism problems for graphic matroids and linear matroids are polynomial time Turing equivalent to the corresponding isomorphism problems. In this context, we give a polynomial time membership test algorithm for the automorphism group of a graphic matroid (Theorem 6.5).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.