On the Parallel Complexity of Group Isomorphism via Weisfeiler-Leman
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2021
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| author | Grochow, Joshua A. Levet, Michael |
| author_facet | Grochow, Joshua A. Levet, Michael |
| contents | 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, Torán, \& 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 $β_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper bound due to Chattopadhyay, Torán, \& Wagner (ibid.). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_11487 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Parallel Complexity of Group Isomorphism via Weisfeiler-Leman Grochow, Joshua A. Levet, Michael Data Structures and Algorithms Computational Complexity Logic in Computer Science Group Theory 20-08, 68Q17, 68Q19, 68Q25, 20A15 F.2.2; I.1.2; F.4.1 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, Torán, \& 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 $β_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper bound due to Chattopadhyay, Torán, \& Wagner (ibid.). |
| title | On the Parallel Complexity of Group Isomorphism via Weisfeiler-Leman |
| topic | Data Structures and Algorithms Computational Complexity Logic in Computer Science Group Theory 20-08, 68Q17, 68Q19, 68Q25, 20A15 F.2.2; I.1.2; F.4.1 |
| url | https://arxiv.org/abs/2112.11487 |