Journal of Combinatorial Optimization

, Volume 8, Issue 3, pp 329–361 | Cite as

Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results

  • Clemens Heuberger


Given a (combinatorial) optimization problem and a feasible solution to it, the corresponding inverse optimization problem is to find a minimal adjustment of the cost function such that the given solution becomes optimum.

Several such problems have been studied in the last twelve years. After formalizing the notion of an inverse problem and its variants, we present various methods for solving them. Then we discuss the problems considered in the literature and the results that have been obtained. Finally, we formulate some open problems.

inverse optimization reverse optimization network flow problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R.K. Ahuja, Th. L. Magnanti, and J.B. Orlin, Network Flows. Theory, Algorithms, and Applications, Prentice, Hall Inc.: Englewood Cliffs, NJ, 1993.Google Scholar
  2. R.K. Ahuja and J.B. Orlin, “A faster algorithm for the inverse spanning tree problem,” J. Algorithms, vol. 34, pp. 177–193, 2000.CrossRefGoogle Scholar
  3. R.K. Ahuja and J.B. Orlin, “Inverse optimization,” Oper. Res., vol. 49, pp. 771–783, 2001.CrossRefGoogle Scholar
  4. R.K. Ahuja and J.B. Orlin, “Combinatorial algorithms for inverse network flow problems,” Networks, vol. 40, no. 4, pp. 181–187, 2002.CrossRefGoogle Scholar
  5. C. Berge, “Two theorems in graph theory,” Proc. Nat. Acad. Sci. U.S.A., vol. 43, pp. 842–844, 1957.Google Scholar
  6. O. Berman, D.I. Ingco, and A. Odoni, “Improving the location of minisum facilities through network modification,” Ann. Oper. Res., vol. 40, pp. 1–16, 1992.Google Scholar
  7. O. Berman, D.I. Ingco, and A. Odoni, “Improving the location of minimax facilities through network modification,” Networks, vol. 24, pp. 31–41, 1994.Google Scholar
  8. R.E. Burkard, B. Klinz, and J. Zhang, “Bottleneck capacity expansion problems with general budget constraints,” RAIRO Oper. Res., vol. 35, pp. 1–20, 2001.CrossRefGoogle Scholar
  9. R.E. Burkard, Y. Lin, and J. Zhang, “Weight reduction problems with certain bottleneck objectives,” European J. Oper. Res., vol. 153, no. 1, pp. 191–199, 2004a.CrossRefGoogle Scholar
  10. R.E. Burkard, C. Pleschiutschnig, and J. Zhang, “Inverse median problems,” Discrete Optimization, vol.1, pp. 23–39, 2004b.CrossRefGoogle Scholar
  11. D. Burton, “On the inverse shortest path problem,” Doctoral dissertation, Facultés Universitaires Notre-Dame de la Paix de Namur, Dápartement de Mathématique, Namur, Belgium, 1993. Available at Scholar
  12. D. Burton and Ph. L. Toint, “On an instance of the inverse shortest paths problem,” Math. Programming, vol. 53, pp. 45–61, 1992.Google Scholar
  13. D. Burton and Ph. L. Toint, “On the use of an inverse shortest paths algorithm for recovering linearly correlated costs,” Math. Programming, vol. 63, pp. 1–22, 1992.Google Scholar
  14. D. Burton, W.R. Pulleyblank, and Ph. L. Toint, “The inverse shortest paths problem with upper bounds on shortest paths costs,” in Network optimization Gainesville, FL, 1996, pp. 156–171. Springer: Berlin, 1997.Google Scholar
  15. M. C. Cai, “Inverse problems of matroid intersection,” J. Comb. Optim., vol. 3, pp. 465–474, 1999.CrossRefGoogle Scholar
  16. M.C. Cai and Y. Li, “Inverse matroid intersection problem,” Math. Methods Oper. Res., vol. 45, pp. 235–243, 1997.Google Scholar
  17. M.C. Cai and X.G. Yang, “Inverse shortest path problems,” in Operations Research and its Applications. First International Symposium, ISORA '95, Beijing, P.R. China, August 19-22, 1995. D.Z. Du, X.S. Zhang, and K. Cheng (eds.), Proceedings of Lecture Notes in Operations Research, BeijingWorld Publishing Corporation, 1995, vol. 1, pp. 242–248.Google Scholar
  18. M.C. Cai, X.G. Yang, and Y. Li, “Inverse problems of submodular functions on digraphs,” J. Optim. Theory Appl., vol. 104, pp. 559–575, 2000.CrossRefGoogle Scholar
  19. M.C. Cai, X.G. Yang, and Y. Li, “Inverse polymatroidal flow problem,” J. Comb. Optim., vol. 3, pp. 115–126, 1999.CrossRefGoogle Scholar
  20. M.C. Cai, X.G. Yang, and J. Zhang, “The complexity analysis of the inverse center location problem,” J. Global Optim., vol. 15, pp. 213–218, 1999.CrossRefGoogle Scholar
  21. M. Dell'Amico, F. Maffioli, and F. Malucelli, “The base-matroid and inverse combinatorial optimization problems,” Discrete Appl. Math., vol. 128, pp. 337–353, 2003.CrossRefGoogle Scholar
  22. R.B. Dial, “Minimal-revenue congestion pricing Part I: A fast algorithm for the single-origin case,” Transportation Res.Part B, vol. 33, pp. 189–202, 1999.CrossRefGoogle Scholar
  23. R.B. Dial, “Minimal-revenue congestion pricing Part II:Anefficient algorithm for the general case,” Transportation Res.Part B, vol. 34, pp. 645–665, 2000.CrossRefGoogle Scholar
  24. K.U. Drangmeister, S.O. Krumke, M.V. Marathe, H. Noltemeier, and S.S. Ravi, “Modifying edges of a network to obtain short subgraphs,” Theoret. Comput. Sci., vol. 203, pp. 91–121, 1998.CrossRefGoogle Scholar
  25. J. Edmonds and R. Giles, “A min-max relation for submodular functions on graphs,” in Studies in Integer Programming (Proc. Workshop, Bonn, 1975), of Ann. of Discrete Math., North-Holland: Amsterdam, 1977, vol. 1, pp. 185–204.Google Scholar
  26. H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group: Dordrecht, 1996.Google Scholar
  27. A. Frank, “A weighted matroid intersection algorithm,” J. Algorithms, vol. 2, pp. 328–336, 1981.Google Scholar
  28. S.P. Fekete, W. Hochstättler, St. Kromberg, and Ch. Moll, “The complexity of an inverse shortest paths problem,” in Contemporary trends in discrete mathematics ( ? Sti?rín Castle, 1997), Amer. Math. Soc., Providence, RI, 1999, pp. 113–127.Google Scholar
  29. A. Frank, “An algorithm for submodular functions on graphs,” in Bonn Workshop on Combinatorial Optimization (Bonn, 1980), North-Holland: Amsterdam, 1982, pp. 97–120.Google Scholar
  30. A. Frank, “Packing paths, circuits, and cuts-a survey,” in Paths, flows, and VLSI-layout (Bonn, 1988), Springer, Berlin, 1990, pp. 47–100.Google Scholar
  31. G.N. Frederickson and R. Solis-Oba, “Efficient algorithms for robustness in matroid optimization,” in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997), ACM: New York, 1997, pp. 659–668.Google Scholar
  32. G.N. Frederickson and R. Solis-Oba, “Increasing the weight of minimum spanning trees,” J. Algorithms, vol. 33, pp. 244–266, 1999.CrossRefGoogle Scholar
  33. D.R. Fulkerson and G.C. Harding, “Maximizing the minimum source-sink path subject to a budget constraint,” Math. Programming, vol. 13, pp. 116–118, 1977.Google Scholar
  34. D. Goldfarb and A. Idnani, “A numerically stable dual method for solving strictly convex quadratic programs,” Math. Programming, vol. 27, pp. 1–33, 1983.Google Scholar
  35. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, 2nd edn. Springer-Verlag: Berlin, 1993.Google Scholar
  36. S. Hakimi, “Optimum locations of switching centers and the absolute centers and medians of a graph,” Oper. Res., vol. 12, pp. 450–459, 1964.Google Scholar
  37. R. Hassin, “Minimum cost flow with set-constraints,” Networks, vol. 12, pp. 1–21, 1982.Google Scholar
  38. D.S. Hochbaum, “Efficient algorithms for the inverse spanning-tree problem,” Oper. Res., vol. 51, no. 5, pp. 785–797, 2003.CrossRefGoogle Scholar
  39. Z. Hu and Z. Liu, “A strongly polynomial algorithm for the inverse shortest arborescence problem,” Discrete Appl. Math., vol. 82, pp. 135–154, 1998.CrossRefGoogle Scholar
  40. S. Huang and Z. Liu, “On the inverse problem of linear programming and its application to minimum weight perfect k-matching,” Eur. J. Oper. Res., vol. 112, pp. 421–426, 1999.CrossRefGoogle Scholar
  41. S.O. Krumke, M.V.Marathe, H. Noltemeier, R.Ravi, and S.S. Ravi, “Approximation algorithms for certain network improvement problems,” J. Comb. Optim., vol. 2, pp. 257–288, 1998.CrossRefGoogle Scholar
  42. E.L. Lawler and C.U. Martel, “Computing maximal “polymatroidal” network flows,” Math. Oper. Res., vol. 7, pp. 334–347, 1982.Google Scholar
  43. Z. Liu and J. Zhang, “On inverse problems of optimum perfect matching,” J. Comb. Optim., vol. 7, no. 3, pp. 215–228, 2003.CrossRefGoogle Scholar
  44. B. Marlow, “Inverse problems,” Scholar
  45. T.J. Moser, “Shortest paths calculation of seismic rays,” Geophysics, vol. 56, pp. 59–67, 1991.CrossRefGoogle Scholar
  46. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons Inc.: New York, 1988.Google Scholar
  47. C. Phillips, “The network inhibition problem,” in Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, 1993, San Diego, CA USA, May 16-18, 1993, pp. 776–785.Google Scholar
  48. T. Radzik, “Parametric flows, weighted means of cuts, and fractional combinatorial optimization,” in Complexity in Numerical Optimization, World Sci. Publishing: River Edge, NJ, 1993, pp. 351–386.Google Scholar
  49. A. Schrijver, “Total dual integrality from directed graphs, crossing families, and sub-and supermodular functions,” in Progress in Combinatorial OptimizationWaterloo, Ont., 1982, Academic Press: Toronto, Ont., 1984, pp. 315–361.Google Scholar
  50. A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons Ltd.: Chichester, 1986.Google Scholar
  51. P.T. Sokkalingam, “The minimum cost flow problem: Primal algorithms and cost perturbations,” Unpublished dissertation, Department of Mathematics, Indian Institute of Technology, Kanpur, India, 1995.Google Scholar
  52. P.T. Sokkalingam, R.K. Ahuja, and J.B. Orlin, “Solving inverse spanning tree problems through network flow techniques,” Oper. Res., vol. 47, pp. 291–298, 1999.Google Scholar
  53. É Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,” Oper. Res., vol. 34, pp. 250–256, 1986.Google Scholar
  54. Q.Wei, J. Zhang, and X. Zhang, “An inverse DEA model for inputs/outputs estimate,” Eur. J. Oper. Res., vol. 121, pp. 151–163, 2000.CrossRefGoogle Scholar
  55. W.L. Winston, Operations Research: Applications and Algorithms, 3rd edn. Duxbury Press: Boston, MA, 1993.Google Scholar
  56. S. Xu and J. Zhang, “An inverse problem of the weighted shortest path problem,” Japan J. Indust. Appl. Math., vol. 12, pp. 47–59, 1995.Google Scholar
  57. C. Yang, “Some inverse optimization problems and extensions,” PhD Thesis, Department of Mathematics, City University of Hong Kong, 1997.Google Scholar
  58. X. Yang, “Complexity of partial inverse assignment problem and partial inverse cut problem,” RAIRO Oper. Res., vol.35, no. 1, pp. 117–126, 2001.CrossRefGoogle Scholar
  59. C. Yang and J. Zhang, “A constrained capacity expansion problem on networks,” Int. J. Comput. Math., vol. 70, pp. 19–33, 1998a.Google Scholar
  60. C. Yang and J. Zhang, “Inverse maximum capacity problems,” OR Spektrum, vol. 20, pp. 97–100, 1998b.Google Scholar
  61. C. Yang and J. Zhang, “Two general methods for inverse optimization problems,” Appl. Math. Lett., vol. 12, pp. 69–72, 1999.CrossRefGoogle Scholar
  62. C. Yang, J. Zhang, and Z. Ma, “Inverse maximum flow and minimum cut problems,” Optimization, vol. 40, pp. 147–170, 1997.Google Scholar
  63. J. Zhang and M.C. Cai, “Inverse problem of minimum cuts,” Math. Methods Oper. Res., vol. 47, pp. 51–58, 1998.Google Scholar
  64. J. Zhang and Z. Liu, “Calculating some inverse linear programming problems,”J. Comput. Appl. Math., vol. 72, pp. 261–273, 1996.CrossRefGoogle Scholar
  65. J. Zhang and Z. Liu, “A further study on inverse linear programming problems,”J. Comput. Appl. Math., vol. 106, pp. 345–359, 1999.CrossRefGoogle Scholar
  66. J. Zhang and Z. Liu, “A general model of some inverse optimization problems and its solution method under l8 norm,” J. Comb. Optim., vol. 6, pp. 207–227, 2002a.CrossRefGoogle Scholar
  67. J. Zhang and Z. Liu, “An oracle strongly polynomial algorithm for bottleneck expansion problems,” Optimization Methods and Software, vol. 17, pp. 61–75, 2002b.CrossRefGoogle Scholar
  68. J. Zhang and Z. Ma, “A network flow method for solving some inverse combinatorial optimization problems,” Optimization, vol. 37, pp. 59–72, 1996.Google Scholar
  69. J. Zhang and Z. Ma, “Solution structure of some inverse combinatorial optimization problems,” J. Comb. Optim., vol. 3, pp. 127–139, 1999a.CrossRefGoogle Scholar
  70. J. Zhang, Z. Ma, and C. Yang, “A column generation method for inverse shortest path problems,” ZOR-Math. Methods Oper. Res., vol. 41, pp. 347–358, 1995.Google Scholar
  71. J. Zhang, Z. Liu, and Z. Ma, “On the inverse problem of minimum spanning tree with partition constraints,” Math. Methods Oper. Res., vol. 44, pp. 171–187, 1996.Google Scholar
  72. J. Zhang, S. Xu, and Z. Ma, “An algorithm for inverse minimum spanning tree problem,” Optim. Methods Softw., vol. 8, pp. 69–84, 1997.Google Scholar
  73. J. Zhang, Z. Liu, and Z. Ma, “The inverse fractional matching problem,” J. Austral. Math. Soc. Ser. B, vol. 40, pp. 484–496, 1999.Google Scholar
  74. J. Zhang, X.G. Yang, and M.C. Cai, “Reverse center location problem,” in A. Aggarwal and C. Pandu Rangan, editors, Algorithms and computation. 10th International Symposium, ISAAC' 99, Chennai, India, December 16-18, 1999. Proceedings, vol. 1741 of Lecture Notes in Computer Science, Springer, 1999b, pp. 279–294.Google Scholar
  75. J. Zhang, Z. Liu, and Z. Ma, “Some reverse location problems,” Eur. J. Oper. Res., vol. 124, pp. 77–88, 2000.CrossRefGoogle Scholar
  76. J. Zhang, C. Yang, and Y. Lin, “A class of bottleneck expansion problems,” Computers and Operations Research, vol. 28, pp. 505–519, 2001.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Clemens Heuberger
    • 1
  1. 1.Institut für Mathematik BTechnische Universität GrazGrazAustria

Personalised recommendations