A Better-Than-1.6-Approximation for Prize-Collecting TSP
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929672512602112 |
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| author | Blauth, Jannis Klein, Nathan Nägele, Martin |
| author_facet | Blauth, Jannis Klein, Nathan Nägele, Martin |
| contents | Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below $1.6$, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of $1.774$. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by $(1+\sqrt{5})/2 \approx 1.618$, the golden ratio. With some additional technical care we further improve it to $1.599$. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06254 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Better-Than-1.6-Approximation for Prize-Collecting TSP Blauth, Jannis Klein, Nathan Nägele, Martin Data Structures and Algorithms Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below $1.6$, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of $1.774$. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by $(1+\sqrt{5})/2 \approx 1.618$, the golden ratio. With some additional technical care we further improve it to $1.599$. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926. |
| title | A Better-Than-1.6-Approximation for Prize-Collecting TSP |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2308.06254 |