Approximating Graphic Multi-Path TSP and Graphic Ordered TSP
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| Format: | Preprint |
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2025
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| _version_ | 1866914015056232448 |
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| author | Alimi, Morteza Dahlmeier, Niklas Mömke, Tobias Pabst, Philipp Koch, Laura Vargas |
| author_facet | Alimi, Morteza Dahlmeier, Niklas Mömke, Tobias Pabst, Philipp Koch, Laura Vargas |
| contents | The path version of the Traveling Salesman Problem is one of the most well-studied variants of the ubiquitous TSP. Its generalization, the Multi-Path TSP, has recently been used in the best known algorithm for path TSP by Traub and Vygen [Cambridge University Press, 2024]. The best known approximation factor for this problem is $2.214$ by Böhm, Friggstad, Mömke and Spoerhase [SODA 2025]. In this paper we show that for the case of graphic metrics, a significantly better approximation guarantee of $2$ can be attained. Our algorithm is based on sampling paths from a decomposition of the flow corresponding to the optimal solution to the LP for the problem, and connecting the left-out vertices with doubled edges. The cost of the latter is twice the optimum in the worst case; we show how the cost of the sampled paths can be absorbed into it without increasing the approximation factor. Furthermore, we prove that any below-$2$ approximation algorithm for the special case of the problem where each source is the same as the corresponding sink yields a below-$2$ approximation algorithm for Graphic Multi-Path TSP.
We also show that our ideas can be utilized to give a factor $1.791$-approximation algorithm for Ordered TSP in graphic metrics, for which the aforementioned paper [SODA 2025] and Armbruster, Mnich and Nägele [APPROX 2024] give a $1.868$-approximation algorithm in general metrics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_00448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximating Graphic Multi-Path TSP and Graphic Ordered TSP Alimi, Morteza Dahlmeier, Niklas Mömke, Tobias Pabst, Philipp Koch, Laura Vargas Data Structures and Algorithms 68R05 (Primary), 90C27 (Secondary) F.2.2; G.2.1; E.1 The path version of the Traveling Salesman Problem is one of the most well-studied variants of the ubiquitous TSP. Its generalization, the Multi-Path TSP, has recently been used in the best known algorithm for path TSP by Traub and Vygen [Cambridge University Press, 2024]. The best known approximation factor for this problem is $2.214$ by Böhm, Friggstad, Mömke and Spoerhase [SODA 2025]. In this paper we show that for the case of graphic metrics, a significantly better approximation guarantee of $2$ can be attained. Our algorithm is based on sampling paths from a decomposition of the flow corresponding to the optimal solution to the LP for the problem, and connecting the left-out vertices with doubled edges. The cost of the latter is twice the optimum in the worst case; we show how the cost of the sampled paths can be absorbed into it without increasing the approximation factor. Furthermore, we prove that any below-$2$ approximation algorithm for the special case of the problem where each source is the same as the corresponding sink yields a below-$2$ approximation algorithm for Graphic Multi-Path TSP. We also show that our ideas can be utilized to give a factor $1.791$-approximation algorithm for Ordered TSP in graphic metrics, for which the aforementioned paper [SODA 2025] and Armbruster, Mnich and Nägele [APPROX 2024] give a $1.868$-approximation algorithm in general metrics. |
| title | Approximating Graphic Multi-Path TSP and Graphic Ordered TSP |
| topic | Data Structures and Algorithms 68R05 (Primary), 90C27 (Secondary) F.2.2; G.2.1; E.1 |
| url | https://arxiv.org/abs/2509.00448 |