Count-Free Weisfeiler--Leman and Group Isomorphism
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929218631237632 |
|---|---|
| author | Collins, Nathaniel A. Levet, Michael |
| author_facet | Collins, Nathaniel A. Levet, Michael |
| contents | We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler--Leman Version I algorithm for groups (Brachter & Schweitzer, LICS 2020) in tandem with limited non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include:
- Direct products of non-Abelian simple groups.
- Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an $O(1)$-generated solvable group with solvability class $\text{poly} \log \log n$. This notably includes instances where the complement is an $O(1)$-generated nilpotent group. This problem was previously known to be in $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011), and the complexity was recently improved to $\textsf{L}$ (Grochow & Levet, FCT 2023).
- Graphical groups of class $2$ and exponent $p > 2$ (Mekler, J. Symb. Log., 1981) arising from the CFI and twisted CFI graphs (Cai, Fürer, & Immerman, Combinatorica 1992) respectively. In particular, our work improves upon previous results of Brachter & Schweitzer (LICS 2020).
We finally show that the $q$-ary count-free pebble game is unable to distinguish even Abelian groups. This extends the result of Grochow & Levet (ibid), who established the result in the case of $q = 1$. The general theme is that some counting appears necessary to place Group Isomorphism into $\textsf{P}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Count-Free Weisfeiler--Leman and Group Isomorphism Collins, Nathaniel 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 We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler--Leman Version I algorithm for groups (Brachter & Schweitzer, LICS 2020) in tandem with limited non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: - Direct products of non-Abelian simple groups. - Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an $O(1)$-generated solvable group with solvability class $\text{poly} \log \log n$. This notably includes instances where the complement is an $O(1)$-generated nilpotent group. This problem was previously known to be in $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011), and the complexity was recently improved to $\textsf{L}$ (Grochow & Levet, FCT 2023). - Graphical groups of class $2$ and exponent $p > 2$ (Mekler, J. Symb. Log., 1981) arising from the CFI and twisted CFI graphs (Cai, Fürer, & Immerman, Combinatorica 1992) respectively. In particular, our work improves upon previous results of Brachter & Schweitzer (LICS 2020). We finally show that the $q$-ary count-free pebble game is unable to distinguish even Abelian groups. This extends the result of Grochow & Levet (ibid), who established the result in the case of $q = 1$. The general theme is that some counting appears necessary to place Group Isomorphism into $\textsf{P}$. |
| title | Count-Free Weisfeiler--Leman and Group Isomorphism |
| 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/2212.11247 |