Advertisement

Inverse Maximum Flow Problem Under the Combination of the Weighted l\(_2\) Norm and the Weighted Hamming Distance

  • Long-Cheng LiuEmail author
  • Han Gao
  • Chao Li
Article
  • 2 Downloads

Abstract

The idea of the inverse optimization problem is to adjust the values of the parameters so that the observed feasible solutions are indeed optimal. The modification cost is measured by different norms, such as \(l_1, l_2, l_\infty \) norms and the Hamming distance, and the goal is to adjust the parameters as little as possible. In this paper, we consider the inverse maximum flow problem under the combination of the weighted \(l_2\) norm and the weighted Hamming distance, i.e., the modification cost is fixed in a given interval and depends on the modification out of the given interval. We present a combinatorial algorithm which can be finished in O(nm) to solve it due to the minimum cut of the residual network.

Keywords

Maximum flow Minimum cut Inverse problem Residual network Strongly polynomial algorithm 

Mathematics Subject Classification

05C21 68Q25 90B10 90C27 

Notes

Acknowledgements

The authors wish to thank the anonymous referees whose valuable comments allowed us to improve the paper.

References

  1. 1.
    Ahuja, R.K., Magnant, T.L., Orlin, J.B.: Network Flows: Theory. Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Yang, C., Zhang, J.Z., Ma, Z.F.: Inverse maximum flow and minimum cut problems. Optimization 40, 147–170 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Liu, L.C., Zhang, J.Z.: Inverse maximum flow problems under the weighted Hamming distance. J. Comb. Optim. 12, 395–408 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Deaconu, A.: The inverse maximum flow problem with lower and upper bounds for the flow. Yugosl. J. Oper. Res. 18, 13–22 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Deaconu, A.: The inverse maximum flow problem considering \(l_\infty \) norm. RAIRO Oper. Res. 42, 401–414 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Deaconu, A., Ciurea, E.: The inverse maximum flow problem under \(L_k\) norms. Carpathian J. Math. 28, 59–66 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ciurea, E., Deaconu, A.: Inverse minimum flow problem. J. Appl. Math. Comput. 23, 193–203 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Güler, C., Hamacher, H.W.: Capacity inverse minimum cost flow problem. J. Comb. Optim. 19, 43–59 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tayyebi, J., Aman, M.: Note on “Inverse minimum cost flow problems under the weighted Hamming distance”. Eur. J. Oper. Res. 234, 916–920 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Alizadeh, B., Burkard, R.E., Pferschy, U.: Inverse 1-center location problems with edge length augmentation on trees. Computing 86, 331–343 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guan, X.C., Zhang, B.W.: Inverse 1-median problem on trees under weighted Hamming distance. J. Glob. Optim. 54, 75–82 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nguyen, K.T., Sepasian, A.R.: The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance. J. Comb. Optim. 32, 872–884 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nguyen, K.T., Vui, P.T.: The inverse \(p\)-maxian problem on trees with variable edge lengths. Taiwan. J. Math. 20, 1437–1449 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    He, Y., Zhang, B.W., Yao, E.Y.: Weighted inverse minimum spanning tree problems under Hamming distance. J. Comb. Optim. 9, 91–100 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, L.C., Wang, Q.: Constrained inverse min-max spanning tree problems under the weighted Hamming distance. J. Glob. Optim. 43, 83–95 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, L.C., Yao, E.Y.: Inverse min-max spanning tree problem under the weighted sum-type Hamming distance. Theor. Comput. Sci. 396, 28–34 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhang, B.W., Zhang, J.Z., He, Y.: Constrained inverse minimum spanning tree problems under the bottleneck-type Hamming distance. J. Glob. Optim. 34, 467–474 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, L.C., Yao, E.Y.: A weighted inverse minimum cut problem under the bottleneck type Hamming distance. Asia Pac. J. Oper. Res. 24, 725–736 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, J.Z., Cai, M.C.: Inverse problem of minimum cuts. Math. Methods Oper. Res. 47, 51–58 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Heuberger, C.: Inverse Optimization: a survey on problems, methods, and results. J. Comb. Optim. 8, 329–361 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Orlin, J.B.: Max flows in \(O(nm)\) time, or better. In: Proceeding of annual ACM symposium on theory of computing, pp. 765–774 (2013)Google Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematics and ComputerWuyi UniversityWuyishanChina

Personalised recommendations