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An Optimal Absolute Approximation Algorithm for Computing k Restricted Shortest Paths

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Computing and Combinatorics (COCOON 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15161))

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Abstract

Given a directed graph with non-negative costs and delays on each edge, the k edge-disjoint restricted shortest path problem (kRSP) is to find k edge-disjoint directed paths connecting a pair of distinct vertices s and t, with the aim of minimizing the total cost subject to a delay constraint. In this paper, we present an absolute approximation algorithm via the LP-rounding technique, which guarantees to find \(k-1\) edge-disjoint paths that strictly satisfy the delay constraint and have a total cost no more than that of an optimum solution. The key observation leading to our approach is that in any basic optimal solution of the linear programming relaxation for kRSP, the underlying graph composed of edges with fractional values forms exactly a cycle. We notably show that the cycle always contains a set of edges that, along with the integral edges from the LP, can compose a desired solution. Lastly, we present a method for rounding such a set of edges from this cycle to eventually obtain the desired solution.

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Acknowledgements

The first and the fourth authors are supported by Beijing Natural Science Foundation Project No. Z220004 and National Natural Science Foundation of China (No. 12131003), and the third author is supported by National Natural Science Foundation of China (No. 12271098).

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Authors and Affiliations

  1. Institute of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Yue Sun & Dachuan Xu

  2. Faculty of Management, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Donglei Du

  3. School of Mathematics and Statistics, Fuzhou University, Fuzhou, 350116, People’s Republic of China

    Longkun Guo

Authors
  1. Yue Sun
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  2. Donglei Du
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  3. Longkun Guo
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  4. Dachuan Xu
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Corresponding author

Correspondence to Longkun Guo .

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Editors and Affiliations

  1. Hangzhou Dianzi University, Hangzhou, Zhejiang, China

    Yong Chen

  2. Shanghai Jiao Tong University, Shanghai, China

    Xiaofeng Gao

  3. Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun

  4. Hangzhou Dianzi University, Hangzhou, Zhejiang, China

    An Zhang

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Sun, Y., Du, D., Guo, L., Xu, D. (2025). An Optimal Absolute Approximation Algorithm for Computing k Restricted Shortest Paths. In: Chen, Y., Gao, X., Sun, X., Zhang, A. (eds) Computing and Combinatorics. COCOON 2024. Lecture Notes in Computer Science, vol 15161. Springer, Singapore. https://doi.org/10.1007/978-981-96-1090-7_2

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  • DOI: https://doi.org/10.1007/978-981-96-1090-7_2

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  • Print ISBN: 978-981-96-1089-1

  • Online ISBN: 978-981-96-1090-7

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