A faster algorithm for Vertex Cover parameterized by solution size
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917071399419904 |
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| author | Harris, David G. Narayanaswamy, N. S. |
| author_facet | Harris, David G. Narayanaswamy, N. S. |
| contents | We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $λ$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover.
The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $μ= k - λ$ are decreased at each step. There can be local obstructions in the graph that prevent $μ$ from decreasing in this process; we develop a number of novel branching steps to handle these situations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_08022 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A faster algorithm for Vertex Cover parameterized by solution size Harris, David G. Narayanaswamy, N. S. Data Structures and Algorithms Combinatorics We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $λ$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $μ= k - λ$ are decreased at each step. There can be local obstructions in the graph that prevent $μ$ from decreasing in this process; we develop a number of novel branching steps to handle these situations. |
| title | A faster algorithm for Vertex Cover parameterized by solution size |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2205.08022 |