Almost-Optimal Approximation Algorithms for Global Minimum Cut in Directed Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914202461929472 |
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| author | Mosenzon, Ron |
| author_facet | Mosenzon, Ron |
| contents | We develop new $(1+ε)$-approximation algorithms for finding the global minimum edge-cut in a directed edge-weighted graph, and for finding the global minimum vertex-cut in a directed vertex-weighted graph. Our algorithms are randomized, and have a running time of $O\left(m^{1+o(1)}/ε\right)$ on any $m$-edge $n$-vertex input graph, assuming all edge/vertex weights are polynomially-bounded. In particular, for any constant $ε>0$, our algorithms have an almost-optimal running time of $O\left(m^{1+o(1)}\right)$. The fastest previously-known running time for this setting, due to (Cen et al., FOCS 2021), is $\tilde{O}\left(\min\left\{n^2/ε^2,m^{1+o(1)}\sqrt{n}\right\}\right)$ for Minimum Edge-Cut, and $\tilde{O}\left(n^2/ε^2\right)$ for Minimum Vertex-Cut. Our results further extend to the rooted variants of the Minimum Edge-Cut and Minimum Vertex-Cut problems, where the algorithm is additionally given a root vertex $r$, and the goal is to find a minimum-weight cut separating any vertex from the root $r$. In terms of techniques, we build upon and extend a framework that was recently introduced by (Chuzhoy et al., SODA 2026) for solving the Minimum Vertex-Cut problem in unweighted directed graphs. Additionally, in order to obtain our result for the Global Minimum Vertex-Cut problem, we develop a novel black-box reduction from this problem to its rooted variant. Prior to our work, such reductions were only known for more restricted settings, such as when all vertex-weights are unit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost-Optimal Approximation Algorithms for Global Minimum Cut in Directed Graphs Mosenzon, Ron Data Structures and Algorithms F.2.2 We develop new $(1+ε)$-approximation algorithms for finding the global minimum edge-cut in a directed edge-weighted graph, and for finding the global minimum vertex-cut in a directed vertex-weighted graph. Our algorithms are randomized, and have a running time of $O\left(m^{1+o(1)}/ε\right)$ on any $m$-edge $n$-vertex input graph, assuming all edge/vertex weights are polynomially-bounded. In particular, for any constant $ε>0$, our algorithms have an almost-optimal running time of $O\left(m^{1+o(1)}\right)$. The fastest previously-known running time for this setting, due to (Cen et al., FOCS 2021), is $\tilde{O}\left(\min\left\{n^2/ε^2,m^{1+o(1)}\sqrt{n}\right\}\right)$ for Minimum Edge-Cut, and $\tilde{O}\left(n^2/ε^2\right)$ for Minimum Vertex-Cut. Our results further extend to the rooted variants of the Minimum Edge-Cut and Minimum Vertex-Cut problems, where the algorithm is additionally given a root vertex $r$, and the goal is to find a minimum-weight cut separating any vertex from the root $r$. In terms of techniques, we build upon and extend a framework that was recently introduced by (Chuzhoy et al., SODA 2026) for solving the Minimum Vertex-Cut problem in unweighted directed graphs. Additionally, in order to obtain our result for the Global Minimum Vertex-Cut problem, we develop a novel black-box reduction from this problem to its rooted variant. Prior to our work, such reductions were only known for more restricted settings, such as when all vertex-weights are unit. |
| title | Almost-Optimal Approximation Algorithms for Global Minimum Cut in Directed Graphs |
| topic | Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2512.09080 |