From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866916309185331200 |
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| author | Chekuri, Chandra Jain, Rhea Kulkarni, Shubhang Zheng, Da Wei Zhu, Weihao |
| author_facet | Chekuri, Chandra Jain, Rhea Kulkarni, Shubhang Zheng, Da Wei Zhu, Weihao |
| contents | In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph $G=(V,E)$, a root vertex $r$ and a set $S \subseteq V$ of $k$ terminals. The goal is to find a min-cost subgraph that connects $r$ to each of the terminals. DST admits an $O(\log^2 k/\log \log k)$-approximation in quasi-polynomial time, and an $O(k^ε)$-approximation for any fixed $ε> 0$ in polynomial-time. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [ICALP 2023] obtained a simple and elegant polynomial-time $O(\log k)$-approximation for DST in planar digraphs via Thorup's shortest path separator theorem. We build on their work and obtain several new results on DST and related problems.
- We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in Friggstad and Mousavi [ICALP 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree, Covering Steiner Tree, and the Polymatroid Steiner Tree problems in planar digraphs. All these problems are hard to approximate to within a factor of $Ω(\log^2 n/\log \log n)$ even in trees.
- We prove that the natural cut-based LP relaxation for DST has an integrality gap of $O(\log^2 k)$ in planar graphs. This is in contrast to general graphs where the integrality gap of this LP is known to be $Ω(k)$ and $Ω(n^δ)$ for some fixed $δ> 0$.
- We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the $O(R + \log k)$ approximation of Friggstad and Mousavi [ICALP 2023] when $R= ω(\log^2 k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs Chekuri, Chandra Jain, Rhea Kulkarni, Shubhang Zheng, Da Wei Zhu, Weihao Data Structures and Algorithms In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph $G=(V,E)$, a root vertex $r$ and a set $S \subseteq V$ of $k$ terminals. The goal is to find a min-cost subgraph that connects $r$ to each of the terminals. DST admits an $O(\log^2 k/\log \log k)$-approximation in quasi-polynomial time, and an $O(k^ε)$-approximation for any fixed $ε> 0$ in polynomial-time. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [ICALP 2023] obtained a simple and elegant polynomial-time $O(\log k)$-approximation for DST in planar digraphs via Thorup's shortest path separator theorem. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in Friggstad and Mousavi [ICALP 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree, Covering Steiner Tree, and the Polymatroid Steiner Tree problems in planar digraphs. All these problems are hard to approximate to within a factor of $Ω(\log^2 n/\log \log n)$ even in trees. - We prove that the natural cut-based LP relaxation for DST has an integrality gap of $O(\log^2 k)$ in planar graphs. This is in contrast to general graphs where the integrality gap of this LP is known to be $Ω(k)$ and $Ω(n^δ)$ for some fixed $δ> 0$. - We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the $O(R + \log k)$ approximation of Friggstad and Mousavi [ICALP 2023] when $R= ω(\log^2 k)$. |
| title | From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.01904 |