Document

# Approximation Algorithms for Covering Vertices by Long Paths

## File

LIPIcs.MFCS.2022.53.pdf
• Filesize: 0.74 MB
• 14 pages

## Cite As

Mingyang Gong, Jing Fan, Guohui Lin, and Eiji Miyano. Approximation Algorithms for Covering Vertices by Long Paths. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.53

## Abstract

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least k vertices is considered long. When k ≤ 3, the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed k ≥ 4, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when k = 4, the problem admits a 4-approximation algorithm which was presented recently. We propose the first (0.4394 k + O(1))-approximation algorithm for the general problem and an improved 2-approximation algorithm when k = 4. Both algorithms are based on local improvement, and their performance analyses are done via amortization.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
##### Keywords
• Path cover
• k-path
• local improvement
• amortized analysis
• approximation algorithm

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. K. Asdre and S. D. Nikolopoulos. A linear-time algorithm for the k-fixed-endpoint path cover problem on cographs. Networks, 50:231-240, 2007.
2. K. Asdre and S. D. Nikolopoulos. A polynomial solution to the k-fixed-endpoint path cover problem on proper interval graphs. Theoretical Computer Science, 411:967-975, 2010.
3. P. Berman and M. Karpinski. 8/7-approximation algorithm for (1,2)-TSP. In ACM-SIAM Proceedings of the Seventeenth Annual Symposium on Discrete Algorithms (SODA'06), pages 641-648, 2006.
4. Y. Cai, G. Chen, Y. Chen, R. Goebel, G. Lin, L. Liu, and An Zhang. Approximation algorithms for two-machine flow-shop scheduling with a conflict graph. In Proceedings of the 24th International Computing and Combinatorics Conference (COCOON 2018), LNCS 10976, pages 205-217, 2018.
5. Y. Chen, Y. Cai, L. Liu, G. Chen, R. Goebel, G. Lin, B. Su, and A. Zhang. Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph. Journal of Combinatorial Optimization, 2021. Accepted on August 2, 2021.
6. Y. Chen, Z.-Z. Chen, C. Kennedy, G. Lin, Y. Xu, and A. Zhang. Approximation algorithms for the directed path partition problems. In Proceedings of FAW 2021, LNCS 12874, pages 23-36, 2021.
7. Y. Chen, R. Goebel, G. Lin, L. Liu, B. Su, W. Tong, Y. Xu, and A. Zhang. A local search 4/3-approximation algorithm for the minimum 3-path partition problem. In Proceedings of FAW 2019, LNCS 11458, pages 14-25, 2019.
8. Y. Chen, R. Goebel, G. Lin, B. Su, Y. Xu, and A. Zhang. An improved approximation algorithm for the minimum 3-path partition problem. Journal of Combinatorial Optimization, 38:150-164, 2019.
9. Y. Chen, R. Goebel, B. Su, W. Tong, Y. Xu, and A. Zhang. A 21/16-approximation for the minimum 3-path partition problem. In Proceedings of ISAAC 2019, LIPIcs 149, pages 46:1-46:20, 2019.
10. H. N. Gabow. An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC'83), pages 448-456, 1983.
11. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Company, San Francisco, 1979.
12. R. Gómez and Y. Wakabayashi. Nontrivial path covers of graphs: Existence, minimization and maximization. Journal of Combinatorial Optimization, 39:437-456, 2020.
13. D. S. Hochbaum. Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics, 6:243-254, 1983.
14. N. Immorlica, M. Mahdian, and V. Mirrokni. Cycle cover with short cycles. In Proceedings of STACS 2005, LNCS 3404, pages 641-653, 2005.
15. K. Kobayashi, G. Lin, E. Miyano, T. Saitoh, A. Suzuki, T. Utashima, and T. Yagita. Path cover problems with length cost. In Proceedings of WALCOM 2022, LNCS 13174, pages 396-408, 2022.
16. J. Monnot and S. Toulouse. The path partition problem and related problems in bipartite graphs. Operations Research Letters, 35:677-684, 2007.
17. M. Neuwohner. An improved approximation algorithm for the maximum weight independent set problem in d-claw free graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), LIPIcs 187, pages 53:1-53:20, 2021.
18. L. L. Pao and C. H. Hong. The two-equal-disjoint path cover problem of matching composition network. Information Processing Letters, 107:18-23, 2008.
19. R. Rizzi, A. I. Tomescu, and V. Mäkinen. On the complexity of minimum path cover with subpath constraints for multi-assembly. BMC Bioinformatics, 15:S5, 2014.
20. J.-H. Yan, G. J. Chang, S. M. Hedetniemi, and S. T. Hedetniemi. k-path partitions in trees. Discrete Applied Mathematics, 78:227-233, 1997.
X

Feedback for Dagstuhl Publishing