Abstract
This paper deals with the problem of finding a collection of vertex-disjoint paths in a given graph \(G=(V,E)\) such that each path has at least four vertices and the total number of vertices in these paths is maximized. The problem is NP-hard and admits an approximation algorithm which achieves a ratio of 2 and runs in \(O(|V|^8)\) time. The known algorithm is based on time-consuming local search, and its authors ask whether one can design a better approximation algorithm by a completely different approach. In this paper, we answer their question in the affirmative by presenting a new approximation algorithm for the problem. Our algorithm achieves a ratio of 1.874 and runs in \(O(\min \{|E|^2|V|^2, |V|^5\})\) time. Unlike the previously best algorithm, ours starts with a maximum matching M of G and then tries to transform M into a solution by utilizing a maximum-weight path-cycle cover in a suitably constructed graph.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Asdre, K., Nikolopoulos, S.D.: A linear-time algorithm for the \(k\)-fixed-endpoint path cover problem on cographs. Networks 50, 231–240 (2007)
Asdre, K., Nikolopoulos, S.D.: A polynomial solution to the \(k\)-fixed-endpoint path cover problem on proper interval graphs. Theoret. Comput. Sci. 411, 967–975 (2010)
Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1,2)-TSP. In: Proceedings of ACM-SIAM SODA 2006, pp. 641–648 (2006)
Cai, Y., et al.: Approximation algorithms for two-machine flow-shop scheduling with a conflict graph. In: Wang, L., Zhu, D. (eds.) COCOON 2018. LNCS, vol. 10976, pp. 205–217. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94776-1_18
Chen, Y., et al.: Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph. J. Comb. Optim. 43, 571–588 (2022)
Chen, Z.-Z., Konno, S., Matsushita, Y.: Approximating maximum edge 2-coloring in simple graphs. Discret. Appl. Math. 158, 1894–1901 (2010)
Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings of ACM STOC 1983, pp. 448–456 (1983)
Gomez, R., Wakabayashi, Y.: Nontrivial path covers of graphs: existence, minimization and maximization. J. Comb. Optim. 39, 437–456 (2020)
Gong, M., Fan, J., Lin, G., Miyano, E.: Approximation algorithms for covering vertices by long paths. In: Proceedings of MFCS 2022. LIPIcs, vol. 241, pp. 53:1–53:14 (2022)
Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discret. Appl. Math. 6, 243–254 (1983)
Kobayashi, K., et al.: Path cover problems with length cost. In: Mutzel, P., Rahman, M.S., Slamin (eds.) WALCOM 2022. LNCS, vol. 13174, pp. 396–408. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-96731-4_32
Kosowski, A.: Approximating the maximum \(2\)- and \(3\)-edge-colorable subgraph problems. Discret. Appl. Math. 157, 3593–3600 (2009)
Micali, S., Vazirani, V.V.: An \({O}(\sqrt{|{V}|} |{E}|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of IEEE FOCS 1980, pp. 17–27 (1980)
Neuwohner, M.: An improved approximation algorithm for the maximum weight independent set problem in \(d\)-claw free graphs. In: Proceedings of STACS 2021, pp. 53:1–53:20 (2021)
Pao, L.L., Hong, C.H.: The two-equal-disjoint path cover problem of matching composition network. Inf. Process. Lett. 107, 18–23 (2008)
Rizzi, R., Tomescu, A.I., Mäkinen, V.: On the complexity of minimum path cover with subpath constraints for multi-assembly. BMC Bioinform. 15, S5 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Gong, M., Chen, ZZ., Lin, G., Wang, L. (2024). An Approximation Algorithm for Covering Vertices by \(4^+\)-Paths. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_33
Download citation
DOI: https://doi.org/10.1007/978-3-031-49611-0_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-49610-3
Online ISBN: 978-3-031-49611-0
eBook Packages: Computer ScienceComputer Science (R0)