On the Parallel Complexity of Group Isomorphism via Weisfeiler-Leman

TL;DR

This paper leverages the Weisfeiler-Leman algorithm to significantly improve parallel complexity bounds for group isomorphism testing across various group families, reducing the complexity class for some cases from P to L.

Contribution

It introduces new parallel algorithms for group isomorphism testing using Weisfeiler-Leman, including for groups with Abelian normal Hall subgroups and direct products, and explores limitations of count-free WL.

Findings

Groups with Abelian normal Hall subgroups are identified in L.

Isomorphism testing for certain groups is in SAC with O(log n) depth.

Count-free WL cannot distinguish Abelian groups in polynomial time.

Abstract

In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show: - Groups with an Abelian normal Hall subgroup whose complement is $O(1)$-generated are identified by constant-dimensional Weisfeiler-Leman using only a constant number of rounds. This places isomorphism testing for this family of groups into $\textsf{L}$; the previous upper bound for isomorphism testing was $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011). - We use the individualize-and-refine paradigm to obtain an isomorphism test for groups without Abelian normal subgroups by $\textsf{SAC}$ circuits of depth $O(\log n)$ and size $n^{O(\log \log n)}$, previously only known to be in $\textsf{P}$ (Babai, Codenotti, \& Qiao, ICALP 2012) and $\mathsf{quasiSAC}^1$ (Chattopadhyay, Tor\'an, \& Wagner, ACM Trans. Comput. Theory, 2013). - We extend a result of Brachter \& Schweitzer (ESA, 2022) on direct products of groups to the parallel setting. Namely, we also show that Weisfeiler--Leman can identify direct products in parallel, provided it can identify each of the indecomposable direct factors in parallel. They previously showed the analogous result for $\textsf{P}$. We finally consider the count-free Weisfeiler--Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of $\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper bound due to Chattopadhyay, Tor\'an, \& Wagner (ibid.).