A Fast Vertex-Swap Operator for the Prize-Collecting Steiner Tree Problem

  • Yi-Fei Ming
  • Si-Bo Chen
  • Yong-Quan Chen
  • Zhang-Hua Fu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10861)


The prize-collecting Steiner tree problem (PCSTP) is one of the important topics in computational science and operations research. The vertex-swap operation, which involves removal and addition of a pair of vertices based on a given minimum spanning tree (MST), has been proven very effective for some particular PCSTP instances with uniform edge costs. This paper extends the vertex-swap operator to make it applicable for solving more general PCSTP instances with varied edge costs. Furthermore, we adopt multiple dynamic data structures, which guarantee that the total time complexity for evaluating all the \(O(n^2)\) possible vertex-swap moves is bounded by \(O(n)\cdot O(m\cdot \,\)log\(\,n)\), where n and m denote the number of vertices and edges respectively (if we run Kruskal’s algorithm with a Fibonacci heap from scratch after swapping any pair of vertices, the total time complexity would reach \(O(n^2) \cdot O(m + n\cdot \,\)log\(\,n)\)). We also prove that after applying the vertex-swap operation, the resulting solutions are necessarily MSTs (unless infeasible).


Computational complexity Network design Prize-collecting Steiner tree Vertex-swap operator Dynamic data structures 



This paper is partially supported by the National Natural Science Foundation of China (grant No: U1613216), the State Joint Engineering Lab on Robotics and Intelligent Manufacturing, and Shenzhen Engineering Lab on Robotics and Intelligent Manufacturing, from Shenzhen Gov, China.


  1. 1.
    Johnson, D.S., Minkoff, M., Phillips, S.: The prize collecting Steiner tree problem: theory and practice. In: Proceeding of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, USA, pp. 760–769 (2000)Google Scholar
  2. 2.
    Canuto, S.A., Resende, M.G.C., Ribeiro, C.C.: Local search with perturbations for the prize collecting Steiner tree problem in graphs. Networks 38, 50–58 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Goldbarg, E.F.G., Goldbarg, M.C., Schmidt, C.C.: A hybrid transgenetic algorithm for the prize collecting Steiner tree problem. J. Univers. Comput. Sci. 14, 2491–2511 (2008)Google Scholar
  4. 4.
    Akhmedov, M., Kwee, I., Montemanni, R.: A divide and conquer matheuristic algorithm for the prize-collecting Steiner tree problem. Comput. Oper. Res. 70, 18–25 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fu, Z.H., Hao, J.K.: Knowledge-guided local search for the prize-collecting Steiner tree problem in graphs. Knowl.-Based Syst. 128, 78–92 (2017)CrossRefGoogle Scholar
  6. 6.
    Fu, Z.H., Hao, J.K.: Swap-vertex based neighborhood for Steiner tree problems. Math. Progr. Comput. 9, 297–320 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26, 362–391 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32, 652–686 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Uchoa, E., Werneck, R.F., Fast local search for Steiner trees in graphs. In: 2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments, ALENEX. pp. 1–10. Society for Industrial and Applied Mathematics (2010)Google Scholar
  10. 10.
    Spira, P.M., Pan, A.: On finding and updating spanning trees and shortest paths. SIAM J. Comput. 4, 375–380 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Das, B., Michael, C.L.: Reconstructing a minimum spanning tree after deletion of any node. Algorithmica 31, 530–547 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Yi-Fei Ming
    • 1
  • Si-Bo Chen
    • 1
  • Yong-Quan Chen
    • 1
  • Zhang-Hua Fu
    • 1
  1. 1.Robotics Laboratory for Logistics Service, Institute of Robotics and Intelligent ManufacturingThe Chinese University of Hong Kong, ShenzhenShenzhenChina

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