Journal of Combinatorial Optimization

, Volume 19, Issue 1, pp 43–59 | Cite as

Capacity inverse minimum cost flow problem



Given a directed graph G=(N,A) with arc capacities u ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector \(\hat{u}\) for the arc set A such that a given feasible flow \(\hat{x}\) is optimal with respect to the modified capacities. Among all capacity vectors \(\hat{u}\) satisfying this condition, we would like to find one with minimum \(\|\hat{u}-u\|\) value.

We consider two distance measures for \(\|\hat{u}-u\|\) , rectilinear (L 1) and Chebyshev (L ) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is \(\mathcal{NP}\) -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.


Inverse problems Network flows Minimum cost flows 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms and applications. Prentice Hall, New York Google Scholar
  2. Ahuja RK, Orlin JB (2001) Inverse optimization. Oper Res 49:771–783 MATHCrossRefMathSciNetGoogle Scholar
  3. Ahuja RK, Orlin JB (2002) Combinatorial algorithms of inverse network flow problems. Networks 40:181–187 MATHCrossRefMathSciNetGoogle Scholar
  4. Ausiello G, Crescenzi P, Gambosi G, Kann V, Marchetti-Spaccamela A, Protasi M (1999) Complexity and approximation: combinatorial optimization problems and their approximability properties. Springer, Berlin MATHGoogle Scholar
  5. Burkard RE, Hamacher HW (1981) Minimal cost flows in regular matroids. Math Program Stud 14: 32–47 MATHMathSciNetGoogle Scholar
  6. Burton D, Toint PL (1992) On an instance of the inverse shortest paths problem. Math Program 53:45–61 MATHCrossRefMathSciNetGoogle Scholar
  7. Burton D, Toint PL (1994) On the use of an inverse shortest paths algorithm for recovering linearly correlated costs. Math Program 63:1–22 CrossRefMathSciNetGoogle Scholar
  8. Caprara A, Fischetti M, Toth P (2000) Algorithms for the set covering problem. Ann Oper Res 89:353–371 CrossRefMathSciNetGoogle Scholar
  9. Demetrescu C, Finocchi I (2003) Combinatorial algorithms for feedback problems in directed graphs. Inf Process Lett 86:129–136 MATHCrossRefMathSciNetGoogle Scholar
  10. Garey MR, Johnson DS (1979) Computers and intractability: a guide to theory of NP-completeness. Freeman, New York MATHGoogle Scholar
  11. Goldfarb D, Grigoriadis MD (1988) A computational comparison of the dinic and network simplex methods for maximum flow. Ann Oper Res 13:83–123 CrossRefMathSciNetGoogle Scholar
  12. Hamacher HW, Küfer K-H (2002) Inverse radiation therapy planning—a multiple objective optimization approach. Discrete Appl Math 118:145–161 MATHCrossRefMathSciNetGoogle Scholar
  13. Heuberger C (2004) Inverse optimization: a survey on problems, methods, and results. J Comb Optim 8:329–361 MATHCrossRefMathSciNetGoogle Scholar
  14. Kann V (1992) On the approximability of NP-complete optimization problems. PhD thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Sweden Google Scholar
  15. Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations, pp 85–103 Google Scholar
  16. Klingman D, Napier A, Stutz J (1974) NETGEN: a program for generating large scale capacitated assignment, transportation, and minimum cost flow network problems. Manage Sci 20:814–820 MATHCrossRefMathSciNetGoogle Scholar
  17. Saab Y (2001) A fast and effective algorithm for the feedback arc set problem. J Heuristics 7:235–250 MATHCrossRefGoogle Scholar
  18. Siek JG, Lee L-Q, Lumsdaine A (2002) The boost graph library: user guide and reference manual. Addison-Wesley, Reading Google Scholar
  19. Yang C, Zhang J, Ma Z (1997) Inverse maximum flow and minimum cut problems. Optimization 40: 147–170 MATHCrossRefMathSciNetGoogle Scholar
  20. Zhang J, Liu L (2006) Inverse maximum flow problems under the weighted hamming distance. J Comb Optim 12:395–408 MATHCrossRefMathSciNetGoogle Scholar
  21. Zhang J, Liu Z (1996) Calculating some inverse linear programming problems. J Comput Appl Math 72:261–273 MATHCrossRefMathSciNetGoogle Scholar
  22. Zhang J, Liu Z (2002) A general model of some inverse combinatorial optimization problems and its solution method under l norm. J Comb Optim 6:207–227 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations