On the Ring Isomorphism & Automorphism Problems

Automorphisms of Finite Rings and Applications to Complexity of Problems

In mathematics, automorphisms of algebraic structures play an important role. Automorphisms capture the symmetries inherent in the structures and many important results have been proved by analyzing the automorphism group of the structure. For example, Galois characterized degree five univariate polynomials f over rationals whose roots can be expressed using radicals (using addition, subtractio...

Automorphisms of Rings and Applications to Complexity

Rings are fundamental mathematical objects with two operations, addition and multiplication, suitably defined. A known way of studying the structure of rings is to consider automorphisms of rings. In my PhD thesis I consider finite dimensional rings represented in terms of their additive basis and study the computational complexity of various automorphism problems of rings in this representatio...

Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs

The graph isomorphism problem belongs to the class of NP problems, and has been conjectured intractable, although probably not NP-complete. However, in the context of chemistry, because molecules are a restricted class of graphs, the problem of graph isomorphism can be solved efficiently (i.e., in polynomial-time). This paper presents the theoretical results that for all molecules, the problems...

On the Complexity of Matroid Isomorphism Problems

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in Σ 2 . In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be Σ 2 -complete and is coNPhard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism and matroid isomorphism a...

On Isomorphism of Simplicial Complexes and Their Related Algebras