Approximating the shortest path problem with scenarios
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866929500928868352 |
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| author | Kasperski, Adam Zielinski, Pawel |
| author_facet | Kasperski, Adam Zielinski, Pawel |
| contents | This paper discusses the shortest path problem in a general directed graph with $n$ nodes and $K$ cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within $Ω(\log^{1-ε} K)$ for any $ε>0$ unless NP$\subseteq \text{DTIME}(n^{\text{polylog} \,n})$ even for arc series-parallel graphs and within $Ω(\log n/\log\log n)$ unless NP$\subseteq \text{ZPTIME}(n^{\log\log n})$ for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of~$K$. In this paper, an $\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_08936 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Approximating the shortest path problem with scenarios Kasperski, Adam Zielinski, Pawel Data Structures and Algorithms This paper discusses the shortest path problem in a general directed graph with $n$ nodes and $K$ cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within $Ω(\log^{1-ε} K)$ for any $ε>0$ unless NP$\subseteq \text{DTIME}(n^{\text{polylog} \,n})$ even for arc series-parallel graphs and within $Ω(\log n/\log\log n)$ unless NP$\subseteq \text{ZPTIME}(n^{\log\log n})$ for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of~$K$. In this paper, an $\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation. |
| title | Approximating the shortest path problem with scenarios |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/1806.08936 |