An improved algorithm for the vertex cover $P_3$ problem on graphs of bounded treewidth
Given a graph $G=(V,E)$ and a positive integer $t\geq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $F\subseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VCP_t$ problem is NP-complete for any integer $t\g...
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| Published in: | Discrete mathematics and theoretical computer science Vol. 21 no. 4; no. Discrete Algorithms |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
04.11.2019
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| Subjects: | |
| ISSN: | 1365-8050, 1365-8050 |
| Online Access: | Get full text |
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| Summary: | Given a graph $G=(V,E)$ and a positive integer $t\geq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $F\subseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VCP_t$ problem is NP-complete for any integer $t\geq2$ and has many applications in real world. Recently, the authors presented a dynamic programming algorithm running in time $4^p\cdot n^{O(1)}$ for the $VCP_3$ problem on $n$-vertex graphs with treewidth $p$. In this paper, we propose an improvement of it and improved the time-complexity to $3^p\cdot n^{O(1)}$. The connected vertex cover $P_3$ ($CVCP_3$) problem is the connected variation of the $VCP_3$ problem where $G[F]$ is required to be connected. Using the Cut\&Count technique, we give a randomized algorithm with runtime $4^p\cdot n^{O(1)}$ for the $CVCP_3$ problem on $n$-vertex graphs with treewidth $p$.
Comment: arXiv admin note: text overlap with arXiv:1103.0534 by other authors |
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| ISSN: | 1365-8050 1365-8050 |
| DOI: | 10.23638/DMTCS-21-4-17 |