Advertisement

A polyhedral approach to the generalized minimum labeling spanning tree problem

  • Thiago Gouveia da SilvaEmail author
  • Serigne Gueye
  • Philippe Michelon
  • Luiz Satoru Ochi
  • Lucídio dos Anjos Formiga Cabral
Original Paper

Abstract

The minimum labeling spanning tree problem (MLSTP) is a combinatorial optimization problem that consists in finding a spanning tree in a simple graph G, in which each edge has one label, by using a minimum number of labels. It is an NP-hard problem that was introduced by Chang and Leu (Inf Process Lett 63(5):277–282, 1997.  https://doi.org/10.1016/S0020-0190(97)00127-0). Chen et al. (Comparison of heuristics for solving the gmlst problem, in: Raghavan, Golden, Wasil (eds) Telecommunications modeling, policy, and technology, Springer, Boston, pp 191–217, 2008) subsequently proposed a generalization of the MLSTP, called the generalized minimum labeling spanning tree problem (GMLSTP), that allows a situation in which multiple labels can be assigned to an edge. Here, we show how the GMLSTP can be expressed as an MLSTP in a multigraph. Both problems have applications in various areas such as computer network design, multimodal transportation network design, and data compression. We propose a new compact binary integer programming model to solve exactly the GMLSTP and analyze the polytope associated with the formulation. The paper introduces new concepts, gives the polytope dimension, and describes five new facet families. The polyhedral comparison results for the studied polytope show that the new model is theoretically superior to current state-of-the-art formulations.

Keywords

Polyhedral study Facets Edge-labeled graph Spanning tree 

Mathematics Subject Classification

90C27 90C10 

Supplementary material

13675_2018_99_MOESM1_ESM.pdf (81 kb)
Supplementary material 1 (pdf 80 KB)

References

  1. Brüggemann T, Monnot J, Woeginger GJ (2003) Local search for the minimum label spanning tree problem with bounded color classes. Oper Res Lett 31(3):195–201CrossRefGoogle Scholar
  2. Captivo M, Clímaco JCN, Pascoal MMB (2009) A mixed integer linear formulation for the minimum label spanning tree problem. Comput Oper Res 36(11):3082–3085.  https://doi.org/10.1016/j.cor.2009.02.003 CrossRefGoogle Scholar
  3. Cerulli R, Fink A, Gentili M, Voß S (2005) Metaheuristics comparison for the minimum labelling spanning tree problem. In: Golden B, Raghavan S, Wasil E (eds) The next wave in computing, optimization, and decision technologies. Operations research/computer science interfaces series, vol 29. Springer, Berlin, pp 93–106.  https://doi.org/10.1007/0-387-23529-9_7 CrossRefGoogle Scholar
  4. Chang R-S, Leu S-J (1997) The minimum labeling spanning trees. Inf Process Lett 63(5):277–282.  https://doi.org/10.1016/S0020-0190(97)00127-0 CrossRefGoogle Scholar
  5. Chen Y, Cornick N, Hall AO, Shajpal R, Silberholz J, Yahav I, Golden BL (2008) Comparison of heuristics for solving the gmlst problem. In: Raghavan S, Golden B, Wasil E (eds) Telecommunications modeling, policy, and technology. Springer, Boston, pp 191–217CrossRefGoogle Scholar
  6. Chwatal AM, Raidl GR (2010) Solving the minimum label spanning tree problem by ant colony optimization. In: Arabnia HR, Hashemi RR, Solo AMG (eds) GEM. CSREA Press, Las Vegas, pp 91–97. ISBN: 1-60132-145-7Google Scholar
  7. Chwatal AM, Raidl GR (2011) Solving the minimum label spanning tree problem by mathematical programming techniques. Adv Oper Res.  https://doi.org/10.1155/2011/14373 Google Scholar
  8. Chwatal AM, Raidl GR, Oberlechner K (2009) Solving an extended minimum label spanning tree problem to compress fingerprint templates. J Math Model Algorithms 8(3):293–334. previous technical report version at https://www.ac.tuwien.ac.at/files/pub/chwatal-08a.pdf
  9. Consoli S, Darby-Dowman K, Mladenovic N, Moreno-Perez JA (2009) Greedy randomized adaptive search and variable neighbourhood search for the minimum labelling spanning tree problem. Eur J Oper Res 196(2):440–449.  https://doi.org/10.1016/j.ejor.2008.03.014 CrossRefGoogle Scholar
  10. Consoli S, Mladenović N, Moreno-Perez JA (2015) Solving the minimum labelling spanning tree problem by intelligent optimization. Appl Soft Comput 28(C):440–452.  https://doi.org/10.1016/j.asoc.2014.12.020 CrossRefGoogle Scholar
  11. Granata D, Cerulli R, ScutellàMG Raiconi A (2013) Maximum flow problems and an np-complete variant on edge-labeled graphs. In: Pardalos P, Du DZ, Graham R (eds) Handbook of combinatorial optimization. Springer, New York, pp 1913–1948CrossRefGoogle Scholar
  12. Krumke SO, Wirth H-C (1998) On the minimum label spanning tree problem. Inf Process Lett 66(2):81–85.  https://doi.org/10.1016/S0020-0190(98)00034-9 CrossRefGoogle Scholar
  13. Van-Nes R (2002) Design of multimodal transport networks: a hierachical approach. PhD thesis. Delft UniversityGoogle Scholar
  14. Wan Y, Chert G, Xu Y (2002) A note on the minimum label spanning tree. Inf Process Lett 84(2):99–101.  https://doi.org/10.1016/S0020-0190(02)00230-2 CrossRefGoogle Scholar
  15. Wolsey LA, Nemhauser GL (2014) Integer and combinatorial optimization. Wiley, New YorkGoogle Scholar
  16. Xiong Y, Golden BL, Wasil EA (2005) A one-parameter genetic algorithm for the minimum labeling spanning tree problem. IEEE Trans Evolut Comput 9(1):55–60CrossRefGoogle Scholar
  17. Xiong Y, Golden BL, Wasil EA (2005) Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem. Oper Res Lett 33(1):77–80CrossRefGoogle Scholar
  18. Xiong Y, Golden BL, Wasil EA (2006) Improved heuristics for the minimum label spanning tree problem. IEEE Trans Evol Comput 10(6):700–703.  https://doi.org/10.1109/TEVC.2006.877147 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies 2018

Authors and Affiliations

  1. 1.IFPB, Instituto Federal de Educação, Ciência e Tecnologia da ParaíbaJoão PessoaBrazil
  2. 2.UFF, Universidade Federal FluminenseNiteróiBrazil
  3. 3.LIA-UAPVUniversité d’AvignonAvignonFrance
  4. 4.UFPB, Universidade Federal da ParaíbaJoão PessoaBrazil

Personalised recommendations