Breaking the Barrier $2^k$ for Subset Feedback Vertex Set in Chordal Graphs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910770757894144 |
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| author | Bai, Tian Xiao, Mingyu |
| author_facet | Bai, Tian Xiao, Mingyu |
| contents | The Subset Feedback Vertex Set problem (SFVS), to delete $k$ vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains NP-hard even in split and chordal graphs, and SFVS in Chordal Graphs (SFVS-C) can be considered as an implicit 3-Hitting Set problem. However, it is not easy to solve SFVS-C faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in $\mathcal{O}^{*}(2^{k})$ time, slightly improving the best result $\mathcal{O}^{*}(2.076^{k})$ for 3-Hitting Set. In this paper, we break the "$2^{k}$-barrier" for SFVS-C by giving an $\mathcal{O}^{*}(1.820^{k})$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04726 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Breaking the Barrier $2^k$ for Subset Feedback Vertex Set in Chordal Graphs Bai, Tian Xiao, Mingyu Data Structures and Algorithms The Subset Feedback Vertex Set problem (SFVS), to delete $k$ vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains NP-hard even in split and chordal graphs, and SFVS in Chordal Graphs (SFVS-C) can be considered as an implicit 3-Hitting Set problem. However, it is not easy to solve SFVS-C faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in $\mathcal{O}^{*}(2^{k})$ time, slightly improving the best result $\mathcal{O}^{*}(2.076^{k})$ for 3-Hitting Set. In this paper, we break the "$2^{k}$-barrier" for SFVS-C by giving an $\mathcal{O}^{*}(1.820^{k})$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method. |
| title | Breaking the Barrier $2^k$ for Subset Feedback Vertex Set in Chordal Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2212.04726 |