Unbreakable Decomposition in Close-to-Linear Time
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917751989207040 |
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| author | Anand, Aditya Lee, Euiwoong Li, Jason Long, Yaowei Saranurak, Thatchaphol |
| author_facet | Anand, Aditya Lee, Euiwoong Li, Jason Long, Yaowei Saranurak, Thatchaphol |
| contents | Unbreakable decomposition, introduced by Cygan et al. (SICOMP'19) and Cygan et al. (TALG'20), has proven to be one of the most powerful tools for parameterized graph cut problems in recent years. Unfortunately, all known constructions require at least $Ω_k\left(mn^2\right)$ time, given an undirected graph with $n$ vertices, $m$ edges, and cut-size parameter $k$. In this work, we show the first close-to-linear time parameterized algorithm that computes an unbreakable decomposition. More precisely, for any $0<ε\leq 1$, our algorithm runs in time $2^{O(\frac{k}ε \log \frac{k}ε)}m^{1 + ε}$ and computes a $(O(k/ε), k)$ unbreakable tree decomposition of $G$, where each bag has adhesion at most $O(k/ε)$.
This immediately opens up possibilities for obtaining close-to-linear time algorithms for numerous problems whose only known solution is based on unbreakable decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unbreakable Decomposition in Close-to-Linear Time Anand, Aditya Lee, Euiwoong Li, Jason Long, Yaowei Saranurak, Thatchaphol Data Structures and Algorithms Unbreakable decomposition, introduced by Cygan et al. (SICOMP'19) and Cygan et al. (TALG'20), has proven to be one of the most powerful tools for parameterized graph cut problems in recent years. Unfortunately, all known constructions require at least $Ω_k\left(mn^2\right)$ time, given an undirected graph with $n$ vertices, $m$ edges, and cut-size parameter $k$. In this work, we show the first close-to-linear time parameterized algorithm that computes an unbreakable decomposition. More precisely, for any $0<ε\leq 1$, our algorithm runs in time $2^{O(\frac{k}ε \log \frac{k}ε)}m^{1 + ε}$ and computes a $(O(k/ε), k)$ unbreakable tree decomposition of $G$, where each bag has adhesion at most $O(k/ε)$. This immediately opens up possibilities for obtaining close-to-linear time algorithms for numerous problems whose only known solution is based on unbreakable decomposition. |
| title | Unbreakable Decomposition in Close-to-Linear Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2408.09368 |