🤖 AI Summary
This work aims to reduce the parallel complexity of group isomorphism testing. For groups possessing an abelian normal Hall subgroup with an $O(1)$-generated complement and no nonabelian normal subgroups, we introduce the first systematic application of constant-dimensional Weisfeiler–Leman (WL) algorithms to group isomorphism. We propose a parallelized individualization-refinement paradigm and develop a parallel WL algorithm for recognizing direct product decompositions. Key contributions include: improving the parallel complexity of several group isomorphism problems from $mathbf{P}$ to $mathbf{L}$ or $mathbf{SAC}^1(O(log n))$; establishing the first upper bound for abelian group isomorphism strictly stronger than $mathbf{TC}^0(mathbf{FOLL})$, namely $eta_1mathbf{MAC}^0(mathbf{FOLL})$; and rigorously proving that counting-free WL algorithms 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 $ extsf{L}$; the previous upper bound for isomorphism testing was $ extsf{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 $ extsf{SAC}$ circuits of depth $O(log n)$ and size $n^{O(log log n)}$, previously only known to be in $ extsf{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 $ extsf{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 $eta_{1} extsf{MAC}^{0}( extsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $ extsf{TC}^{0}( extsf{FOLL})$ upper bound due to Chattopadhyay, Tor'an, &Wagner (ibid.).
