Sparsifying Cayley Graphs on Every Group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913984649625600 |
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| author | Hsieh, Jun-Ting Lee, Daniel Z. Mohanty, Sidhanth Putterman, Aaron Zhang, Rachel Yun |
| author_facet | Hsieh, Jun-Ting Lee, Daniel Z. Mohanty, Sidhanth Putterman, Aaron Zhang, Rachel Yun |
| contents | A classic result in graph theory, due to Batson, Spielman, and Srivastava (STOC 2009) shows that every graph admits a $(1 \pm \varepsilon)$ cut (or spectral) sparsifier which preserves only $O(n / \varepsilon^2)$ reweighted edges. However, when applying this result to \emph{Cayley graphs}, the resulting sparsifier is no longer necessarily a Cayley graph -- it can be an arbitrary subset of edges.
Thus, a recent line of inquiry, and one which has only seen minor progress, asks: for any group $G$, do all Cayley graphs over the group $G$ admit sparsifiers which preserve only $\mathrm{polylog}(|G|)/\varepsilon^2$ many re-weighted generators?
As our primary contribution, we answer this question in the affirmative, presenting a proof of the existence of such Cayley graph spectral sparsifiers, along with an efficient algorithm for finding them. Our algorithm even extends to \emph{directed} Cayley graphs, if we instead ask only for cut sparsification instead of spectral sparsification.
We additionally study the sparsification of linear equations over non-abelian groups. In contrast to the abelian case, we show that for non-abelian valued equations, super-polynomially many linear equations must be preserved in order to approximately preserve the number of satisfied equations for any input. Together with our Cayley graph sparsification result, this provides a formal separation between Cayley graph sparsification and sparsifying linear equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparsifying Cayley Graphs on Every Group Hsieh, Jun-Ting Lee, Daniel Z. Mohanty, Sidhanth Putterman, Aaron Zhang, Rachel Yun Data Structures and Algorithms Combinatorics A classic result in graph theory, due to Batson, Spielman, and Srivastava (STOC 2009) shows that every graph admits a $(1 \pm \varepsilon)$ cut (or spectral) sparsifier which preserves only $O(n / \varepsilon^2)$ reweighted edges. However, when applying this result to \emph{Cayley graphs}, the resulting sparsifier is no longer necessarily a Cayley graph -- it can be an arbitrary subset of edges. Thus, a recent line of inquiry, and one which has only seen minor progress, asks: for any group $G$, do all Cayley graphs over the group $G$ admit sparsifiers which preserve only $\mathrm{polylog}(|G|)/\varepsilon^2$ many re-weighted generators? As our primary contribution, we answer this question in the affirmative, presenting a proof of the existence of such Cayley graph spectral sparsifiers, along with an efficient algorithm for finding them. Our algorithm even extends to \emph{directed} Cayley graphs, if we instead ask only for cut sparsification instead of spectral sparsification. We additionally study the sparsification of linear equations over non-abelian groups. In contrast to the abelian case, we show that for non-abelian valued equations, super-polynomially many linear equations must be preserved in order to approximately preserve the number of satisfied equations for any input. Together with our Cayley graph sparsification result, this provides a formal separation between Cayley graph sparsification and sparsifying linear equations. |
| title | Sparsifying Cayley Graphs on Every Group |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2508.08078 |