Journal of Korean Institute of Industrial Engineers (대한산업공학회지)

Korean Institute of Industrial Engineers (대한산업공학회)
권호 표지
DOI QR코드

DOI QR Code

A Comparison of Group Steiner Tree Formulations

그룹-스타이너-트리 문제의 수학적 모형에 대한 연구

  • 명영수 (단국대학교 경상대학 경영학부)
  • Received : 2011.04.22
  • Accepted : 2011.06.09
  • Published : 2011.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

The group Steiner tree problem is a generalization of the Steiner tree problem that is defined as follows. Given a weighted graph with a family of subsets of nodes, called groups, the problem is to find a minimum weighted tree that contains at least one node in each group. We present some existing and some new formulations for the problem and compare the relaxations of such formulations.

Keywords

References

  1. Chekuri, C., Even, G. and Kortsarz, G. (2006), A greedy approximation algorithm for the group Steiner tree problem, Discrete Applied Mathematics, 154, 15-34. https://doi.org/10.1016/j.dam.2005.07.010
  2. Dror, M., Houari, M. and Chaouachi, J. (2000), Generalized spanning trees, European Journal of Operational Research, 120, 583-592. https://doi.org/10.1016/S0377-2217(99)00006-5
  3. Duin, C. W., Volgenant, A. and Voss, S. (2004), Solving group Steiner tree problems as Steiner problems, European Journal of Operational Research, 154, 323-329. https://doi.org/10.1016/S0377-2217(02)00707-5
  4. Feremans, C., Labbe, M., and Laporte, G. (2001), On generalized minimum spanning tree problem, European Journal of Operational Research, 134, 457-458. https://doi.org/10.1016/S0377-2217(00)00267-8
  5. Feremans, C., Labbe, M., and Laporte, G. (2002), A comparative analysis of several formulations for the generalized minimum spanning tree problem, Networks, 39, 29-34. https://doi.org/10.1002/net.10009
  6. Ferreira, C. E., Filho, F. M., and de Oliveria (2006), Some formulations for the group Steiner tree problem, Discrete Applied Mathematics, 154, 1877-1884. https://doi.org/10.1016/j.dam.2006.03.028
  7. Garg, N., Konjevod, G., and Ravi, R. (2000), A polylogarithmic approximation algorithm for the group Steiner tree problem, Journal of Algorithm, 37, 66-84. https://doi.org/10.1006/jagm.2000.1096
  8. Goemans, M. X. (1994), The Steiner tree polytope and related polyhedra, Mathematical Programming, 63, 157-182. https://doi.org/10.1007/BF01582064
  9. Goemans, M. X. and Myung, Y.-S. (1993), A catalog of Steiner tree formulations, Networks, 23, 19-28. https://doi.org/10.1002/net.3230230104
  10. Golden, B., Raghavan, S., and Stanojevic, D. (2005), Heuristic search for the generalized minimum spanning tree problem, INFORMS Journal on Computing, 17, 290-304. https://doi.org/10.1287/ijoc.1040.0077
  11. Houari, M. and Chaouachi, J. S. (2006), Upper and lower bounding strategies for the generalized spanning tree problem, European Journal of Operational Research, 171, 632-647. https://doi.org/10.1016/j.ejor.2004.07.072
  12. Ihler, E., Reich, P., and Widmayer, G. (1999), Class Steiner trees and VLSI-design, Discrete Applied Mathematics, 154, 173-194.
  13. Myung, Y.-S. (2007), A note on some formulations for the group Steiner tree problem, Working paper, Dankook Univ.
  14. Myung, Y.-S., Lee, C. H., and Tcha, D.W. (1995), On the generalized minimum spanning tree problem, Networks, 26, 231-241. https://doi.org/10.1002/net.3230260407
  15. Polzin, T. and Daneshmand, S. V. (2001), A comparison of Steiner tree relaxaions, Discrete Applied Mathematics, 112, 241-261. https://doi.org/10.1016/S0166-218X(00)00318-8
  16. Pop, P. C., Kern, W., and Still, G. (2006), A new relaxation method for the generalized minimum spanning tree problem, European Journal of Operational Research, 170, 900-908. https://doi.org/10.1016/j.ejor.2004.07.058
  17. Reich, G. and Widmayer, P. (1990), Beyond Steinerʼs problems : a VLSI oriented generalization, Proceedings of Graph-Theoretic Concepts in Computer Science (WG-89), Lecture Notes in Computer Science, 411, Springer, Berlin, 196-210.
  18. Salazar, J. J. (2000), A note on the generalized Steiner tree polytope, Discrete Applied Mathematics, 100, 137-144. https://doi.org/10.1016/S0166-218X(99)00200-0
  19. Wang, Z., Che, C. H., and Lim, A. (2006), Tabu Search for Generalized Minimum Spanning Tree Problem, Lecture Notes in Computer Science, 4099, Springer, Berlin.
  20. Yang, B. and Gillard, P. (2000), The class Steiner minimal tree problem : a lower bound and test problem generation, Acta Informatica, 37, 193-211. https://doi.org/10.1007/s002360000042