Advertisement

Approximation algorithms for capacitated partial inverse maximum spanning tree problem

  • Xianyue Li
  • Zhao ZhangEmail author
  • Ruowang Yang
  • Heping Zhang
  • Ding-Zhu DuEmail author
Article
  • 29 Downloads

Abstract

Given an edge weighted graph, and an acyclic edge set, the goal of the partial inverse maximum spanning tree problem is to modify the weight function as little as possible such that there exists a maximum spanning tree with respect to the new weight function containing the given edge set. In this paper, we consider this problem with capacitated constraint under the \(l_{p}\)-norm, where p is an integer and \(p \in [1,+\,\infty )\). Firstly, we characterize the feasible solutions of this problem. Then, we present a \(\root p \of {k}\)-approximation algorithm for this problem when the weight function can only be decreased, where k is the number of edges in the given edge set. Finally, when the weight function can be either decreased and increased, we propose an approximation algorithm for the general case and analyse its approximation ratio. Moreover, we remark that these algorithms can be generalized under the weighted \(l_{p}\)-norm and the weighted sum Hamming distance.

Keywords

Partial inverse problem Spanning tree Approximation algorithm 

Notes

Acknowledgements

This research work is supported in part by National Numerical Windtunnel Project (No. NNW2019ZT5-B16), NSFC (11571155, 11871256, 11771013, 11531011, 61751303), the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-163), and ZJNSFC (LD19A010001).

References

  1. 1.
    Ahuja, R.K., Orlin, J.B.: A faster algorithm for the inverse spanning tree problem. J. Algorithms 34, 177–193 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ben-Ayed, O., Blair, C.E.: Computational difficulties of bilevel linear programming. Oper. Res. 38(3), 556–560 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)CrossRefGoogle Scholar
  4. 4.
    Cai, M.-C., Duin, C.W., Yang, X., Zhang, J.: The partial inverse minimum spanning tree problem when weight increasing is forbidden. Eur. J. Oper. Res. 188, 348–353 (2008)CrossRefGoogle Scholar
  5. 5.
    Dell’Amico, M., Maffioli, F., Malucelli, F.: The base-matroid and inverse combinatorial optimization problems. Discrete Appl. Math. 128, 337–353 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gassner, E.: The partial inverse minimum cut problem with \(L_1\)-norm is strongly NP-hard. RAIRO Oper. Res. 44, 241–249 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guan, X., He, X., Pardalos, P.M., Zhang, B.: Inverse max + sum spanning tree problem under Hamming distance by modifying the sum-cost vector. J. Glob. Optim. 69(4), 911–925 (2017)CrossRefGoogle Scholar
  8. 8.
    Guan, X., Pardalos, P.M., Zhang, B.: Inverse max + sum spanning tree problem under weighted \(l_1\) norm by modifying the sum-cost vector. Optim. Lett. 12(5), 1065–1077 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guan, X., Pardalos, P.M., Zuo, X.: Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted \(l_\infty \) norm. J. Glob. Optim. 61(1), 165–182 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13, 1194–1217 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    He, Y., Zhang, B., Yao, E.: Weighted inverse minimum spanning tree problems under hamming distance. J. Comb. Optim. 9, 91–100 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hochbaum, D.S.: Efficient algorithms for the inverse spanning-tree problem. Oper. Res. 51(5), 785–797 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hochbaum, D.S., Queyranne, M.: Minimizing a convex cost closure set. SIAM J. Discrete Math. 16(2), 192–207 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lai, T., Orlin, J.: The Complexity of Preprocessing, Research Report of Sloan School of Management, MIT (2003)Google Scholar
  15. 15.
    Li, S., Zhang, Z., Lai, H.-J.: Algorithms for constraint partial inverse matroid problem with weight increase forbidden. Theor. Comput. Sci. 640, 119–124 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, X., Shu, X., Huang, H., Bai, J.: Capacitated partial inverse maximum spanning tree under the weighted Hamming distance. J. Comb. Optim. 38(4), 1005–1018 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, X., Yang, R., Zhang, Z., Zhang, H.: Capacitated partial inverse maximum spanning tree under the weighted \(l_\infty \)-norm, SubmittedGoogle Scholar
  18. 18.
    Li, X., Zhang, Z., Du, D.-Z.: Partial inverse maximum spanning tree in which weight can only be decreased under \(l_p\)-norm. J. Glob. Optim. 30, 677–685 (2018)CrossRefGoogle Scholar
  19. 19.
    Liu, L., Wang, Q.: Constrained inverse min-max spanning tree problems under the weighted Hamming distance. J. Glob. Optim. 43, 83–95 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu, L., Yao, E.: Inverse min–max spanning tree problem under the weighted sum-type Hamming distance. Theor. Comput. Sci. 196, 28–34 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sokkalingam, P.T., Ahuja, R.K., Orlin, J.B.: Solving inverse spanning tree problems through network flow techniques. Oper. Res. 47, 291–298 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yang, X.: Complexity of partial inverse assignment problem and partial inverse cut problem. RAIRO Oper. Res. 35, 117–126 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yang, X., Zhang, J.: Partial inverse assignment problem under \(l_1\) norm. Oper. Res. Lett. 35, 23–28 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yang, X., Zhang, J.: Inverse sorting problem by minimizing the total weighted number of changers and partial inverse sorting problem. Comput. Optim. Appl. 36(1), 55–66 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, B., Zhang, J., He, Y.: Constrained inverse minimum spanning tree problems under the bottleneck-type Hamming distance. J. Glob. Optim. 34, 467–474 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, J., Xu, S., Ma, Z.: An algorithm for inverse minimum spanning tree problem. Optim. Methods Softw. 8(1), 69–84 (1997)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhang, Z., Li, S., Lai, H.-J., Du, D.-Z.: Algorithms for the partial inverse matroid problem in which weights can only be increased. J. Glob. Optim. 65(4), 801–811 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.College of Mathematics and Computer ScienceZhejiang Normal UniversityJinhuaPeople’s Republic of China
  3. 3.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

Personalised recommendations