Optimization and Engineering

, Volume 15, Issue 4, pp 855–887 | Cite as

Cutting-plane-based algorithms for two branch vertices related spanning tree problems

Article

Abstract

A branch vertex is a vertex with degree larger than or equal to three. This paper addresses two spanning tree problems in an undirected, simple graph. The first one is to find a spanning tree that minimizes the number of branch vertices (MBV), and the second one is to find a spanning tree that minimizes the degree sum of branch vertices (MDS). These two problems arise in the design of wavelength-division networks (WDN), when the cost of equipments for enabling multicast communication is to be minimized. After investigating the relations of MBV and MDS with the problem of minimizing the number of leaves in a spanning tree, new formulations based on ILP are proposed for MBV and MDS, along with two cutting plane algorithms for addressing them. A repair function is also introduced for deriving feasible solutions from the candidate trees returned at each iteration of the cutting plane algorithm, as well as a Tabu search procedure for further quality improvement. The resulting hybrid approach is shown to outperform pure ILP formulations and heuristic approaches published earlier.

Keywords

Spanning tree Branch vertex Cutting plane algorithm Matheuristic Tabu search Wavelength-division 

References

  1. Arora A, Subramaniam S (2002) Wavelength conversion placement in WDM mesh optical networks. Photonic Netw Commun 4(2):167–177 CrossRefGoogle Scholar
  2. Cerulli R, Gentili M, Iossa A (2009) Bounded-degree spanning tree problems: models and new algorithms. Comput Optim Appl 42:353–370 CrossRefMathSciNetMATHGoogle Scholar
  3. Diaz I (2006) A branch-and-cut algorithm for graph coloring. Discrete Appl Math 154(5):826–847 CrossRefMathSciNetMATHGoogle Scholar
  4. Dirac G (1952) Some theorems on abstract graphs. Proc Lond Math Soc (3) 2:69–81 CrossRefMathSciNetMATHGoogle Scholar
  5. Elmirghani J, Mouftah H (2000) All-optical wavelength conversion technologies and applications in DWDM networks. IEEE Commun Mag 38(3):86–92 CrossRefGoogle Scholar
  6. Fernandes L, Gouveia L (1998) Minimal spanning trees with a constraint on the number of leaves. Eur J Oper Res 101:250–261 CrossRefGoogle Scholar
  7. FICO (2009) Xpress-MP. http://www.dashoptimization.com/
  8. Gargano L, Hell P, Stacho L, Vaccaro U (2002) Spanning trees with bounded number of branch vertices. In: Lecture notes in computer science, vol 2380. Springer, Berlin, pp 355–363 Google Scholar
  9. Glover F, Laguna M (1997) Tabu search. Kluwer Academic, Norwell CrossRefMATHGoogle Scholar
  10. Graham R, Lawler E, Lenstra J, Rinnooy Kan A (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 4:287–326 CrossRefMathSciNetGoogle Scholar
  11. Jajszczyk A (2005) Optical networks—the electro-optic reality. Opt Switch Netw 1(1):3–18 CrossRefGoogle Scholar
  12. Jia X, Du D, Hu X, Huang H, Li D (2003) On the optimal placement of wavelength converters in WDM networks. Comput Commun 26:986–995 CrossRefGoogle Scholar
  13. Kleinberg J, Kumar A (2001) Wavelength conversion in optical networks. J Algorithms 38(1):25–50 CrossRefMathSciNetMATHGoogle Scholar
  14. Lee C, Kim H (2007) Reliable overlay multicast trees for private Internet broadcasting with multiple sessions. Comput Oper Res 34(9):2849–2864 CrossRefMATHGoogle Scholar
  15. Leggieri V, Nobili P, Triki C (2008) Minimum power multicasting problem in wireless networks. Math Methods Oper Res 68:295–311 CrossRefMathSciNetMATHGoogle Scholar
  16. Montemanni R, Gambardella LM (2005) Exact algorithms for the minimum power symmetric connectivity problem in wireless networks. Comput Oper Res 32:2891–2904 CrossRefMathSciNetMATHGoogle Scholar
  17. Naadef D (2004) Polyhedral theory and branch-and-cut algorithm for the symmetric TSP. In: Gutin G, Punnen A (eds) The traveling salesman problem and its variations. Kluwer Academic, Norwell. Chapter 2 Google Scholar
  18. Pinedo M (2007) Planning and scheduling in manufacturing and services. Springer, New York Google Scholar
  19. Salamon G, Wiener G (2008) On finding spanning trees with few leaves. Inf Process Lett 105:164–169 CrossRefMathSciNetMATHGoogle Scholar
  20. Tomlinson W, Lin C (1978) Optical wavelength-division multiplexer for the 1–1.4-micron spectral region. Electron Lett 14:345–347 CrossRefGoogle Scholar
  21. Volkmann L (1996) Estimation for the number of cycles in a graph. Period Math Hung 33(2):153–161 CrossRefMATHGoogle Scholar
  22. Wilfong G, Winkler P (1998) Ring routing and wavelength transmission. In: Proceedings of the ninth annual ACM-SIAM symposium on discrete algorithms (SODA), pp 333–341 Google Scholar
  23. Wolsey L (1998) Integer programming. Wiley-Interscience, New York MATHGoogle Scholar
  24. Zhou Y, Poo G (2005) Optical multicast over wavelength-routed WDM networks: a survey. Opt Switch Netw 2(3):176–197 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Lab-STICC, CNRSUniversité de Bretagne-SudLorient CedexFrance
  2. 2.Department of Computer and Information SciencesUniversity of HyderabadHyderabadIndia

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