Approximately covering vertices by order-$5$ or longer paths
Abstract
This paper studies $MPC^{5+}_v$, which is to cover as many vertices as possible in a given graph $G=(V,E)$ by vertex-disjoint $5^+$-paths (i.e., paths each with at least five vertices). $MPC^{5+}_v$ is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of $\frac {19}7\approx 2.714$ and runs in $O(|V|^6)$ time. In this paper, we present a new approximation algorithm for $MPC^{5+}_v$ which achieves a ratio of $2.511$ and runs in $O(|V|^{2.5} |E|^2)$ time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.11225
- arXiv:
- arXiv:2408.11225
- Bibcode:
- 2024arXiv240811225G
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Computer Science - Discrete Mathematics
- E-Print:
- Full version of an extended abstract in COCOON 2024