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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.17740 |
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| _version_ | 1866911222023061504 |
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| author | Jiang, Shunhua Kapralov, Michael Li, Lawrence Sidford, Aaron |
| author_facet | Jiang, Shunhua Kapralov, Michael Li, Lawrence Sidford, Aaron |
| contents | In this paper we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \in E$ has a loss factor $γ_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\tilde{O}( (m + n^{1.5}) \cdot \mathrm{polylog}(\frac{W}δ))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $δ$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses. This improves upon the previous state-of-the-art $\tilde{O}(m \sqrt{n} \cdot \log^2(\frac{W}δ) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman, 2008] with techniques from [Lee-Sidford, 2014]. To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang, 2021]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Flow in Nearly-linear Time on Moderately Dense Graphs Jiang, Shunhua Kapralov, Michael Li, Lawrence Sidford, Aaron Data Structures and Algorithms In this paper we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \in E$ has a loss factor $γ_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\tilde{O}( (m + n^{1.5}) \cdot \mathrm{polylog}(\frac{W}δ))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $δ$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses. This improves upon the previous state-of-the-art $\tilde{O}(m \sqrt{n} \cdot \log^2(\frac{W}δ) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman, 2008] with techniques from [Lee-Sidford, 2014]. To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang, 2021]. |
| title | Generalized Flow in Nearly-linear Time on Moderately Dense Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.17740 |