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

Article

- 1.1k Downloads
- 143 Citations

## Abstract

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

## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - 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 - R.K. Ahuja and J.B. Orlin, “Inverse optimization,”
*Oper. Res.*, vol. 49, pp. 771–783, 2001.CrossRefGoogle Scholar - 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 - C. Berge, “Two theorems in graph theory,”
*Proc. Nat. Acad. Sci. U.S.A.*, vol. 43, pp. 842–844, 1957.Google Scholar - 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 - 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 - 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 - 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 - R.E. Burkard, C. Pleschiutschnig, and J. Zhang, “Inverse median problems,”
*Discrete Optimization*, vol.1, pp. 23–39, 2004b.CrossRefGoogle Scholar - 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 http://www.mit.edu/people/dburton/docs/thesis.ps.gz.Google Scholar
- 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 - 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 - 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 - M. C. Cai, “Inverse problems of matroid intersection,”
*J. Comb. Optim.*, vol. 3, pp. 465–474, 1999.CrossRefGoogle Scholar - M.C. Cai and Y. Li, “Inverse matroid intersection problem,”
*Math. Methods Oper. Res.*, vol. 45, pp. 235–243, 1997.Google Scholar - 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 - 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 - M.C. Cai, X.G. Yang, and Y. Li, “Inverse polymatroidal flow problem,”
*J. Comb. Optim.*, vol. 3, pp. 115–126, 1999.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - H.W. Engl, M. Hanke, and A. Neubauer,
*Regularization of Inverse Problems*, Kluwer Academic Publishers Group: Dordrecht, 1996.Google Scholar - A. Frank, “A weighted matroid intersection algorithm,”
*J. Algorithms*, vol. 2, pp. 328–336, 1981.Google Scholar - 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 - 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 - 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 - 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 - G.N. Frederickson and R. Solis-Oba, “Increasing the weight of minimum spanning trees,”
*J. Algorithms*, vol. 33, pp. 244–266, 1999.CrossRefGoogle Scholar - 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 - 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 - M. Grötschel, L. Lovász, and A. Schrijver,
*Geometric Algorithms and Combinatorial Optimization*, 2nd edn. Springer-Verlag: Berlin, 1993.Google Scholar - 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 - R. Hassin, “Minimum cost flow with set-constraints,”
*Networks*, vol. 12, pp. 1–21, 1982.Google Scholar - D.S. Hochbaum, “Efficient algorithms for the inverse spanning-tree problem,”
*Oper. Res*., vol. 51, no. 5, pp. 785–797, 2003.CrossRefGoogle Scholar - 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 - 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 - 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 - E.L. Lawler and C.U. Martel, “Computing maximal “polymatroidal” network flows,”
*Math. Oper. Res.*, vol. 7, pp. 334–347, 1982.Google Scholar - 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 - B. Marlow, “Inverse problems,” http://www.inverse-problems.com/.Google Scholar
- T.J. Moser, “Shortest paths calculation of seismic rays,”
*Geophysics*, vol. 56, pp. 59–67, 1991.CrossRefGoogle Scholar - G.L. Nemhauser and L.A. Wolsey,
*Integer and Combinatorial Optimization*, John Wiley & Sons Inc.: New York, 1988.Google Scholar - 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 - 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 - 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 - A. Schrijver,
*Theory of Linear and Integer Programming*, John Wiley & Sons Ltd.: Chichester, 1986.Google Scholar - 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
- 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 - É Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,”
*Oper. Res.*, vol. 34, pp. 250–256, 1986.Google Scholar - 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 - W.L. Winston, Operations Research: Applications and Algorithms, 3rd edn. Duxbury Press: Boston, MA, 1993.Google Scholar
- 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 - C. Yang, “Some inverse optimization problems and extensions,” PhD Thesis, Department of Mathematics, City University of Hong Kong, 1997.Google Scholar
- 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 - C. Yang and J. Zhang, “A constrained capacity expansion problem on networks,”
*Int. J. Comput. Math.*, vol. 70, pp. 19–33, 1998a.Google Scholar - C. Yang and J. Zhang, “Inverse maximum capacity problems,”
*OR Spektrum*, vol. 20, pp. 97–100, 1998b.Google Scholar - C. Yang and J. Zhang, “Two general methods for inverse optimization problems,”
*Appl. Math. Lett.*, vol. 12, pp. 69–72, 1999.CrossRefGoogle Scholar - C. Yang, J. Zhang, and Z. Ma, “Inverse maximum flow and minimum cut problems,”
*Optimization*, vol. 40, pp. 147–170, 1997.Google Scholar - J. Zhang and M.C. Cai, “Inverse problem of minimum cuts,”
*Math. Methods Oper. Res.*, vol. 47, pp. 51–58, 1998.Google Scholar - J. Zhang and Z. Liu, “Calculating some inverse linear programming problems,”
*J. Comput. Appl. Math.*, vol. 72, pp. 261–273, 1996.CrossRefGoogle Scholar - 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 - J. Zhang and Z. Liu, “A general model of some inverse optimization problems and its solution method under
*l*8 norm,”*J. Comb. Optim.*, vol. 6, pp. 207–227, 2002a.CrossRefGoogle Scholar - 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 - 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 - J. Zhang and Z. Ma, “Solution structure of some inverse combinatorial optimization problems,”
*J. Comb. Optim.*, vol. 3, pp. 127–139, 1999a.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - J. Zhang, Z. Liu, and Z. Ma, “Some reverse location problems,”
*Eur. J. Oper. Res.*, vol. 124, pp. 77–88, 2000.CrossRefGoogle Scholar - 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