On the Complexity of Matroid Isomorphism Problem

TL;DR

This paper investigates the computational complexity of matroid isomorphism problems, establishing polynomial-time equivalences with graph isomorphism and providing algorithms for automorphism testing, especially for graphic and bounded rank linear matroids.

Contribution

It demonstrates polynomial-time equivalences between matroid isomorphism, graphic matroid isomorphism, and graph isomorphism, and provides new algorithms for automorphism group membership testing.

Findings

Matroid isomorphism is in $ ext{NP}$ and related to graph isomorphism.

Polynomial-time reductions connect matroid isomorphism with graph isomorphism.

Automorphism group membership testing for graphic matroids is polynomial-time solvable.

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 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.