The Module Isomorphism Problem Reconsidered

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -- such as groups, and almost any type of algebra -- seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a MAGMA implementation of our algorithm

A Heuristic for Direct Product Graph Decomposition

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph G with adjacency matrix Adj(G), our heuristic aims at identifying two graphs G 1 and G 2 such that G = G 1 × G 2 , where G 1 × G 2 is the direct product of G 1 and G 2 . For undirected, connected graphs it has been shown that graph decomposition is “at least as difficult” as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents – to the best of our knowledge – the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably- sized instances in a few seconds on general-purpose hardware. Although the proposed heuristic is not guaranteed to find a factorization, even if one exists; however, it always succeeds on all the randomly-generated instances used in the experimental evaluation

On the Complexity of Polytope Isomorphism Problems

We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard. The original version of the paper (June 2001, 11 pages) had the title ``On the Complexity of Isomorphism Problems Related to Polytopes''. The main difference between the current and the former version is a new polynomial time algorithm for polytope isomorphism in bounded dimension that does not rely on Luks polynomial time algorithm for checking two graphs of bounded valence for isomorphism. Furthermore, the treatment of geometric isomorphism problems was extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the Complexity of Isomorphism Problems Related to Polytopes'' (June 2001

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A