Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911323763245056 |
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| author | Jiang, Yonggang Long, Yaowei Saranurak, Thatchaphol Wang, Benyu |
| author_facet | Jiang, Yonggang Long, Yaowei Saranurak, Thatchaphol Wang, Benyu |
| contents | We give almost-linear-time algorithms for approximating rooted minimum cut and maximum arborescence packing in directed graphs, two problems that are dual to each other [Edm73]. More specifically, for an $n$-vertex, $m$-edge directed graph $G$ whose $s$-rooted minimum cut value is $k$, our first algorithm computes an $s$-rooted cut of size at most $O(k\log^{5} n)$ in $m^{1+o(1)}$ time, and our second algorithm packs $k$ $s$-rooted arborescences with $n^{o(1)}$ congestion in $m^{1+o(1)}$ time, certifying that the $s$-rooted minimum cut is at least $k / n^{o(1)}$. Our first algorithm also works for weighted graphs.
Prior to our work, the fastest algorithms for computing the $s$-rooted minimum cut were exact but had super-linear running time: either $\tilde{O}(mk)$ [Gab91] or $\tilde{O}(m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\})$ [CLN+22]. The fastest known algorithms for packing $s$-rooted arborescences had no congestion, but required $\tilde{O}(m \cdot \mathrm{poly}(k))$ time [BHKP08]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies Jiang, Yonggang Long, Yaowei Saranurak, Thatchaphol Wang, Benyu Data Structures and Algorithms We give almost-linear-time algorithms for approximating rooted minimum cut and maximum arborescence packing in directed graphs, two problems that are dual to each other [Edm73]. More specifically, for an $n$-vertex, $m$-edge directed graph $G$ whose $s$-rooted minimum cut value is $k$, our first algorithm computes an $s$-rooted cut of size at most $O(k\log^{5} n)$ in $m^{1+o(1)}$ time, and our second algorithm packs $k$ $s$-rooted arborescences with $n^{o(1)}$ congestion in $m^{1+o(1)}$ time, certifying that the $s$-rooted minimum cut is at least $k / n^{o(1)}$. Our first algorithm also works for weighted graphs. Prior to our work, the fastest algorithms for computing the $s$-rooted minimum cut were exact but had super-linear running time: either $\tilde{O}(mk)$ [Gab91] or $\tilde{O}(m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\})$ [CLN+22]. The fastest known algorithms for packing $s$-rooted arborescences had no congestion, but required $\tilde{O}(m \cdot \mathrm{poly}(k))$ time [BHKP08]. |
| title | Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2512.05300 |