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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2308.14727 |
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| _version_ | 1866929403009695744 |
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| author | Dong, Sally Ye, Guanghao |
| author_facet | Dong, Sally Ye, Guanghao |
| contents | We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $τ$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrtτ + S)$ time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver by [Dong-Lee-Ye,21] and [Gu-Song,22], which runs in $\widetilde{O}(m τ^{(ω+1)/2})$ time, where $ω\approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). For general graphs where treewidth is trivially bounded by $n$, the algorithm runs in $\widetilde{O}(m \sqrt n)$ time, which is the best-known result without using the Lee-Sidford barrier or $\ell_1$ IPM, demonstrating the surprising power of robust interior point methods.
As a corollary, we obtain a $\widetilde{O}(\operatorname{tw}^3 \cdot m)$ time algorithm to compute a tree decomposition of width $O(\operatorname{tw}\cdot \log(n))$, given a graph with $m$ edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs Dong, Sally Ye, Guanghao Data Structures and Algorithms We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $τ$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrtτ + S)$ time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver by [Dong-Lee-Ye,21] and [Gu-Song,22], which runs in $\widetilde{O}(m τ^{(ω+1)/2})$ time, where $ω\approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). For general graphs where treewidth is trivially bounded by $n$, the algorithm runs in $\widetilde{O}(m \sqrt n)$ time, which is the best-known result without using the Lee-Sidford barrier or $\ell_1$ IPM, demonstrating the surprising power of robust interior point methods. As a corollary, we obtain a $\widetilde{O}(\operatorname{tw}^3 \cdot m)$ time algorithm to compute a tree decomposition of width $O(\operatorname{tw}\cdot \log(n))$, given a graph with $m$ edges. |
| title | Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2308.14727 |