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An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance

Author

Listed:
  • Binwu Zhang

    (Hohai University)

  • Xiucui Guan

    (Southeast University)

  • Panos M. Pardalos

    (University of Florida
    Higher School of Economics)

  • Chunyuan He

    (Hohai University)

Abstract

Shortest path problems play important roles in computer science, communication networks, and transportation networks. In a shortest path improvement problem under unit Hamming distance, an edge-weighted graph with a set of source–terminal pairs is given. The objective is to modify the weights of the edges at a minimum cost under unit Hamming distance such that the modified distances of the shortest paths between some given sources and terminals are upper bounded by the given values. As the shortest path improvement problem is NP-hard, it is meaningful to analyze the complexity of the shortest path improvement problem defined on rooted trees with one common source. We first present a preprocessing algorithm to normalize the problem. We then present the proofs of some properties of the optimal solutions to the problem. A dynamic programming algorithm is proposed for the problem, and its time complexity is analyzed. A comparison of the computational experiments of the dynamic programming algorithm and MATLAB functions shows that the algorithm is efficient although its worst-case complexity is exponential time.

Suggested Citation

  • Binwu Zhang & Xiucui Guan & Panos M. Pardalos & Chunyuan He, 2018. "An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 538-559, August.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:2:d:10.1007_s10957-018-1221-9
    DOI: 10.1007/s10957-018-1221-9
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    References listed on IDEAS

    as
    1. Jiang, Yiwei & Liu, Longcheng & Wu, Biao & Yao, Enyu, 2010. "Inverse minimum cost flow problems under the weighted Hamming distance," European Journal of Operational Research, Elsevier, vol. 207(1), pages 50-54, November.
    2. Yong He & Binwu Zhang & Enyu Yao, 2005. "Weighted Inverse Minimum Spanning Tree Problems Under Hamming Distance," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 91-100, February.
    3. Binwu Zhang & Jianzhong Zhang & Liqun Qi, 2006. "The shortest path improvement problems under Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 12(4), pages 351-361, December.
    4. Xiucui Guan & Xinyan He & Panos M. Pardalos & Binwu Zhang, 2017. "Inverse max $$+$$ + sum spanning tree problem under Hamming distance by modifying the sum-cost vector," Journal of Global Optimization, Springer, vol. 69(4), pages 911-925, December.
    5. Duin, C.W. & Volgenant, A., 2006. "Some inverse optimization problems under the Hamming distance," European Journal of Operational Research, Elsevier, vol. 170(3), pages 887-899, May.
    6. Xiucui Guan & Binwu Zhang, 2012. "Inverse 1-median problem on trees under weighted Hamming distance," Journal of Global Optimization, Springer, vol. 54(1), pages 75-82, September.
    7. Binwu Zhang & Jianzhong Zhang & Yong He, 2005. "The Center Location Improvement Problem Under the Hamming Distance," Journal of Combinatorial Optimization, Springer, vol. 9(2), pages 187-198, March.
    8. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
    Full references (including those not matched with items on IDEAS)

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