Linear algebraic analogues of the graph isomorphism problem and the Erd\H{o}s-R\'enyi model

TL;DR

This paper introduces a linear algebraic approach to the graph isomorphism problem, applying it to group isomorphism and developing average-case algorithms inspired by classical graph techniques, with implications for understanding automorphism groups.

Contribution

It proposes a linear algebraic analogue of graph isomorphism, develops an average-case algorithm for the matrix space isometry problem, and adapts classical graph techniques to this algebraic setting.

Findings

An average-case efficient algorithm for the matrix space isometry problem.

Most graphs have trivial automorphism groups, extended to a linear algebraic setting.

Luks' dynamic programming technique improves worst-case complexity in certain parameters.

Abstract

A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpoint leads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of group isomorphism. In 1970's, Babai, Erd\H{o}s, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, we devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters, in a random model of alternating matrix spaces in vein of the Erd\H{o}s-R\'enyi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. As a consequence of the main algorithm, we establish a weaker linear algebraic analogue of Erd\H{o}s and R\'enyi's classical result that most graphs have the trivial automorphism group. We finally show that Luks' dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.