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Solution Structure of Some Inverse Combinatorial Optimization Problems

Author

Listed:
  • Jianzhong Zhang

    (City University of Hong Kong)

  • Zhongfan Ma

    (Academia Sinica)

Abstract

In this paper we consider some inverse combinatorial optimization problems, that is, for a given set of feasible solutions of an optimization problem, we wish to know: under what weight vectors (or capacity vectors) will these feasible solutions become optimal solutions? We analysed shortest path problem, minimum cut problem, minimum spanning tree problem and maximum-weight matching problem, and found that in each of these cases, the solution set of the inverse problem is charaterized by solving another combinatorial optimization problem. The main tool in our approach is Fulkerson's theory of blocking and anti-blocking polyhedra with some necessary revisions.

Suggested Citation

  • Jianzhong Zhang & Zhongfan Ma, 1999. "Solution Structure of Some Inverse Combinatorial Optimization Problems," Journal of Combinatorial Optimization, Springer, vol. 3(1), pages 127-139, July.
    • Handle: RePEc:spr:jcomop:v:3:y:1999:i:1:d:10.1023_a:1009829525096
    DOI: 10.1023/A:1009829525096
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    References listed on IDEAS

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    1. Cai Mao-Cheng & Yanjun Li, 1997. "Inverse Matroid Intersection Problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(2), pages 235-243, June.
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    Cited by:

    1. Yi Zhang & Liwei Zhang & Yue Wu, 2014. "The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 45-79, April.
    2. Zhang, Jianzhong & Liu, Zhenhong & Ma, Zhongfan, 2000. "Some reverse location problems," European Journal of Operational Research, Elsevier, vol. 124(1), pages 77-88, July.
    3. Jianzhong Zhang & Liwei Zhang & Xiantao Xiao, 2010. "A Perturbation approach for an inverse quadratic programming problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 379-404, December.
    4. Jia Wu & Yi Zhang & Liwei Zhang & Yue Lu, 2016. "A Sequential Convex Program Approach to an Inverse Linear Semidefinite Programming Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(04), pages 1-26, August.
    5. Chen, Lu & Chen, Yuyi & Langevin, André, 2021. "An inverse optimization approach for a capacitated vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 295(3), pages 1087-1098.
    6. M. Cai & X. Yang & Y. Li, 2000. "Inverse Problems of Submodular Functions on Digraphs," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 559-575, March.
    7. Yong He & Binwu Zhang & Enyu Yao, 2005. "Weighted Inverse Minimum Spanning Tree Problems Under Hamming Distance," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 91-100, February.
    8. Wei, Quanling & Zhang, Jianzhong & Zhang, Xiangsun, 2000. "An inverse DEA model for inputs/outputs estimate," European Journal of Operational Research, Elsevier, vol. 121(1), pages 151-163, February.
    9. Zhenhong Liu & Jianzhong Zhang, 2003. "On Inverse Problems of Optimum Perfect Matching," Journal of Combinatorial Optimization, Springer, vol. 7(3), pages 215-228, September.
    10. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.

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