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

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, Erdős, 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 Erdős-Rényi 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 Erdős and Rényi’s classical result that most graphs have the trivial automorphism group. We also 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. Most notable progress on the worst-case time complexity of graph isomorphism, including Babai’s recent breakthrough (STOC 2016) and Babai and Luks’ previous record (STOC 1983), has relied on both group theoretic and combinatorial techniques. By developing a linear algebraic analogue of the individualisation technique and demonstrating its usefulness in the average-case setting, the main result opens up the possibility of adapting that strategy for graph isomorphism to this hard instance of group isomorphism. The linear algebraic Erdős-Rényi model is of independent interest and may deserve further study. In particular, we indicate a connection with enumerating p-groups of class 2 and exponent p. ∗Centre for Quantum Software and Information, University of Technology Sydney, Australia ([email protected]). †Centre for Quantum Software and Information, University of Technology Sydney, Australia ([email protected]). ar X iv :1 70 8. 04 50 1v 2 [ cs .D S] 1 O ct 2 01 7