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Generalized Steiner Problems and Other Variants

Author

Listed:
  • Moshe Dror

    (The University of Arizona)

  • Mohamed Haouari

    (Ecole Polytechnique de Tunisie)

Abstract

In this paper, we examine combinatorial optimization problems by considering the case where the set N (the ground set of elements) is expressed as a union of a finite number of m nonempty distinct subsets N 1,...,N m. The term we use is the generalized Steiner problems coined after the Generalized Traveling Salesman Problem. We have collected a short list of classical combinatorial optimization problems and we have recast each of these problems in this broader framework in an attempt to identify a linkage between these “generalized” problems. In the literature one finds generalized problems such as the Generalized Minimum Spanning Tree (GMST), Generalized Traveling Salesman Problem (GTSP) and Subset Bin-packing (SBP). Casting these problems into the new problem setting has important implications in terms of the time effort required to compute an optimal solution or a “good” solution to a problem. We examine questions like “is the GTSP “harder” than the TSP?” for a number of paradigmatic problems starting with “easy” problems such as the Minimal Spanning Tree, Assignment Problem, Chinese Postman, Two-machine Flow Shop, and followed by “hard” problems such as the Bin-packing, and the TSP.

Suggested Citation

  • Moshe Dror & Mohamed Haouari, 2000. "Generalized Steiner Problems and Other Variants," Journal of Combinatorial Optimization, Springer, vol. 4(4), pages 415-436, December.
    • Handle: RePEc:spr:jcomop:v:4:y:2000:i:4:d:10.1023_a:1009881326671
    DOI: 10.1023/A:1009881326671
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    References listed on IDEAS

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    1. Olivier Goldschmidt & David Nehme & Gang Yu, 1994. "Note: On the set‐union knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(6), pages 833-842, October.
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    6. Dror, M. & Haouari, M. & Chaouachi, J., 2000. "Generalized spanning trees," European Journal of Operational Research, Elsevier, vol. 120(3), pages 583-592, February.
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    Cited by:

    1. Volgenant, A., 2004. "Solving the k-cardinality assignment problem by transformation," European Journal of Operational Research, Elsevier, vol. 157(2), pages 322-331, September.
    2. Pop, Petrică C. & Cosma, Ovidiu & Sabo, Cosmin & Sitar, Corina Pop, 2024. "A comprehensive survey on the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 314(3), pages 819-835.
    3. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2003. "Generalized network design problems," European Journal of Operational Research, Elsevier, vol. 148(1), pages 1-13, July.
    4. W Zahrouni & H Kamoun, 2011. "Transforming part-sequencing problems in a robotic cell into a GTSP," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 114-123, January.
    5. Snyder, Lawrence V. & Daskin, Mark S., 2006. "A random-key genetic algorithm for the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 174(1), pages 38-53, October.
    6. Masoumeh Zojaji & Mohammad Reza Mollakhalili Meybodi & Kamal Mirzaie, 0. "A rapid learning automata-based approach for generalized minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-24.
    7. Masoumeh Zojaji & Mohammad Reza Mollakhalili Meybodi & Kamal Mirzaie, 2020. "A rapid learning automata-based approach for generalized minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 40(3), pages 636-659, October.
    8. Pop, Petrica C. & Kern, W. & Still, G., 2006. "A new relaxation method for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 170(3), pages 900-908, May.
    9. Pop, Petrică C., 2020. "The generalized minimum spanning tree problem: An overview of formulations, solution procedures and latest advances," European Journal of Operational Research, Elsevier, vol. 283(1), pages 1-15.
    10. Mehdi El Krari & Belaïd Ahiod & Youssef Bouazza El Benani, 2021. "A pre-processing reduction method for the generalized travelling salesman problem," Operational Research, Springer, vol. 21(4), pages 2543-2591, December.

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