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An \(O(n(m+n\log n)\log n)\) time algorithm to solve the minimum cost tension problem

  • Mehdi Ghiyasvand
Article
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Abstract

This paper presents an \(O(n(m+n\log n)\log n)\) time algorithm to solve the minimum cost tension problem, where n and m denote the number of nodes and number of arcs, respectively. The algorithm is inspired by Orlin (Oper Res 41:338–350, 1993) and improves upon the previous best strongly polynomial time of \(O(\max \{m^3n, m^2\log n(m+n\log n)\})\) due to Ghiyasvand (J Comb Optim 34:203–217, 2017).

Keywords

Network flows Minimum cost tension problem Strongly polynomial time 

Notes

Acknowledgements

I would like to express great appreciation to the editor and three anonymous reviewers for their valuable comments and suggestions, which have helped to improve the quality and presentation of this paper.

References

  1. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  2. Ahuja RK, Hochbaum DS, Orlin JB (2003) Solving the convex cost integer dual network flow problem. Manage Sci 49:950–964CrossRefzbMATHGoogle Scholar
  3. Bachelet B, Duhamel C (2009) Aggregation approach for the minimum binary cost tension problem. Eur J Oper Res 197:837–841CrossRefzbMATHGoogle Scholar
  4. Bachelet B, Mahey P (2003) Minimum convex-cost tension problems on series-parallel graphs. RAIRO Oper Res 37(4):221–234MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bachelet B, Mahey P (2004) Minimum convex piecewise linear cost tension problem on quasi-k series-parallel graphs. 4OR Q J Eur Oper Res Soc 2(4):275–291MathSciNetCrossRefzbMATHGoogle Scholar
  6. Berge C, Ghouila-Houri A (1962) Programming, games and transportation networks. Wiley, New YorkGoogle Scholar
  7. Edmonds I, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. J Assoc Comput Mach 19:248–264CrossRefzbMATHGoogle Scholar
  8. Fredman ML, Tarjan RE (1987) Fibonacci heaps and their uses in improved network optimization algorithms. J Assoc Comput Mach 34:596–615MathSciNetCrossRefGoogle Scholar
  9. Gabow HN (1985) Scaling algorithms for network problems. J Comput Syst Sci 31:148–168MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ghiyasvand M (2012) An \(O(m(m+n\log n)\log (nC))\)-time algorithm to solve the minimum cost tension problem. Theor Comput Sci 448:47–55MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ghiyasvand M (2012) A polynomial-time implementation of Pla’s method to solve the MCT problem. Adv Comput Math Its Appl 1(2):104–109Google Scholar
  12. Ghiyasvand M (2014) Scaling implementation of a tension rectification algorithm to solve the feasible differential problem. Sci Iran 21:980–987Google Scholar
  13. Ghiyasvand M (2017) A faster strongly polynomial time algorithm to solve the minimum cost tension problem. J Comb Optim 34:203–217MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ghouila-Houri A (1964) Flots et tension dans un graphe, Ph.D Thesis, Gauthier-Villars, ParisGoogle Scholar
  15. Goldberg AV, Tarjan RE (1990) Finding minimum-cost circulations by successive approximation. Math Oper Res 16:430–466MathSciNetCrossRefzbMATHGoogle Scholar
  16. Guler C (2008) Inverse tension problems and monotropic optimization. WIMA ReportGoogle Scholar
  17. Hadjiat M (1998) Penelope’s graph: a hard minimum cost tension instance. Theor Comput Sci 194:207–218MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hadjiat M, Maurras JF (1997) A strongly polynomial algorithm for the minimum cost tension problem. Discrete Math 165(166):377–394MathSciNetCrossRefzbMATHGoogle Scholar
  19. Hamacher HW (1985) Min cost tension. J Inf Optim Sci 6(3):285–304MathSciNetzbMATHGoogle Scholar
  20. Karzanov A, McCormick ST (1997) Polynomial methods for separable convex optimization in unimodular linear spaces with applications. SIAM J Comput 26:1245–1275MathSciNetCrossRefzbMATHGoogle Scholar
  21. Maurras JF (1994) The maximum cost tension problem. In: Proceedings Conference, European chapter on combinatorial optimization (ECCO VII), ItalyGoogle Scholar
  22. Orlin JB (1993) A faster strongly polynomial minimum cost flow algorithm. Oper Res 41:338–350MathSciNetCrossRefzbMATHGoogle Scholar
  23. Pla JM (1971) An out-of-kilter algorithm for solving minimum cost potential problems. Math Program 1:275–290MathSciNetCrossRefzbMATHGoogle Scholar
  24. Rockafeller RT (1984) Network flows and monotropic optimization. Wiley, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceBu-Ali Sina UniversityHamedanIran

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