Abstract
The Min-Min problem of finding a disjoint path pair with the length of the shorter path minimized is known to be NP-complete and no K-approximation exists for any K ≥ 1 [1]. In this paper, we give a simpler proof of this result in general digraphs. We show that this proof can be extended to the problem in planar digraphs whose complexity was unknown previously. As a by-product, we show this problem remains NP-complete even when all edge costs are equal (i.e. strongly NP-complete).
This project was partially supported by the “100 Talents” Project of Chinese Academy of China and NSFC grant #622307. The corresponding author is Hong Shen.
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References
Xu, D., Chen, Y., Xiong, Y., Qiao, C., He, X.: On the complexity of and algorithms for finding the shortest path with a disjoint counterpart. IEEE/ACM Transactions on Networking 14(1), 147–158 (2006)
Zheng, S., Yang, B., Yang, M., Wang, J.: Finding Minimum-Cost Paths with Minimum Sharability. In: 26th IEEE International Conference on Computer Communications IEEE INFOCOM 2007, pp. 1532–1540 (2007)
Li, C., Thomas McCormick, S., Simchi-Levi, D.: Finding disjoint paths with different path-costs: Complexity and algorithms. Networks 22(7) (1992)
Bhatia, R., Kodialam, M., Lakshman, T.: Finding disjoint paths with related path costs. Journal of Combinatorial Optimization 12(1), 83–96 (2006)
Suurballe, J.: Disjoint paths in a network. Networks 4(2) (1974)
Suurballe, J., Tarjan, R.: A quick method for finding shortest pairs of disjoint paths. Networks 14(2) (1984)
Li, C., McCormick, T., Simich-Levi, D.: The complexity of finding two disjoint paths with min-max objective function. Discrete Applied Mathematics 26(1), 105–115 (1989)
Seymour, P.: Disjoint paths in graphs. Discrete Mathematics 306(10-11), 979–991 (2006)
Shiloach, Y.: A polynomial solution to the undirected two paths problem. Journal of the ACM (JACM) 27(3), 445–456 (1980)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem* 1. Theoretical Computer Science 10(2), 111–121 (1980)
van der Holst, H., de Pina, J.: Length-bounded disjoint paths in planar graphs. Discrete Applied Mathematics 120(1-3), 251–261 (2002)
Ding, G., Schrijver, A., Seymour, P.: Disjoint Paths in a Planar Graph . SIAM J. Discrete Math. 5(1), 112–116 (1992)
Schrijver, A.: Finding k disjoint paths in a directed planar graph . SIAM J. Computing 23(4), 780–788 (1994)
Guo, L., Shen, H.: On Finding Min-Min disjoint paths (manuscript)
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Guo, L., Shen, H. (2011). Hardness of Finding Two Edge-Disjoint Min-Min Paths in Digraphs. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_32
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DOI: https://doi.org/10.1007/978-3-642-21204-8_32
Publisher Name: Springer, Berlin, Heidelberg
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