[Paper Review] On the Group and Color Isomorphism Problems.

François Le Gall, David J. Rosenbaum|arXiv (Cornell University)|Sep 27, 2016

Complexity and Algorithms in Graphs25 references3 citations

TL;DR

This paper establishes a tight connection between group isomorphism and a restricted color isomorphism problem where composition factors are excluded from the alternating group. It shows that group isomorphism reduces to this color isomorphism problem in n^(O(log log n)) time, and identifies the projective special linear group as the primary obstacle to faster algorithms, paralleling the role of the alternating group in graph isomorphism.

ABSTRACT

In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bou nded degree graph isomorphism, and have therefore focused on the alternating composit ion factors, since those are the bottleneck in the case of graph isomorphism. We consider the color isomorphism problem with composition factors restricted to those other than the alternating group, show that group isomorphism reduces in n^(O(log log n)) time to this problem, and, conversely, that a special case of this color isomorphism problem reduces to a slight generalization of group isomorphism. We then sharpen our results by identifying the projective special linear group as the main obstacle to faster algorithms for group isomorphism and prove that the aforementioned reduc tion from group isomorphism to color isomorphism in fact produces only cyclic and projective special linear factors. Our results demonstrate that, just as the alternatin g group was a barrier to faster algorithms for graph isomorphism for three decades, the projective special linear group is an obstacle to faster algorithms for group isomorphism.

Motivation & Objective

• To understand the complexity relationship between group isomorphism and color isomorphism problems.
• To identify the composition factors that hinder faster algorithms for group isomorphism.
• To show that group isomorphism reduces to a color isomorphism problem excluding alternating group composition factors.
• To demonstrate that the projective special linear group is the central obstacle in this reduction.

Proposed method

• Reduction of group isomorphism to a color isomorphism problem with composition factors restricted to non-alternating groups.
• Analysis of the structure of permutation groups involved, focusing on composition factors beyond the alternating group.
• Identification of the projective special linear group as the dominant factor in the reduction process.
• Proof that the reduction produces only cyclic and projective special linear group factors.
• Use of group-theoretic techniques to analyze the complexity of isomorphism problems.
• Establishment of a bidirectional reduction between a special case of color isomorphism and a generalized group isomorphism problem.

Experimental results

Research questions

• RQ1 What is the complexity relationship between group isomorphism and color isomorphism when alternating group composition factors are excluded?
• RQ2 Why does the projective special linear group act as a bottleneck in group isomorphism algorithms?
• RQ3 Can group isomorphism be reduced to a restricted color isomorphism problem in sub-exponential time?
• RQ4 What are the structural properties of the permutation groups arising in such reductions?
• RQ5 How do the results for group isomorphism compare to the historical role of the alternating group in graph isomorphism?

Key findings

• Group isomorphism reduces to the restricted color isomorphism problem in n^(O(log log n)) time when alternating group composition factors are excluded.
• The reduction produces only cyclic and projective special linear group factors, identifying the latter as the primary obstacle.
• A special case of the restricted color isomorphism problem reduces to a generalized group isomorphism problem.
• The projective special linear group plays a role analogous to the alternating group in graph isomorphism, acting as a barrier to faster algorithms.
• The results establish a tight structural and complexity connection between group isomorphism and color isomorphism under restricted composition factors.
• The findings suggest that progress on group isomorphism algorithms hinges on overcoming challenges posed by the projective special linear group.