Abstract
We consider the inverse maximum dynamic flow (IMDF) problem. IMDF problem can be described as: how to change the capacity vector of a dynamic network as little as possible so that a given feasible dynamic flow becomes a maximum dynamic flow. After discussing some characteristics of this problem, it is converted to a constrained minimum dynamic cut problem. Then an efficient algorithm which uses two maximum dynamic flow algorithms is proposed to solve the problem.
Similar content being viewed by others
References
Ahuja R K, Orlin J B. Inverse optimization. Oper Res, 2001, 49: 771–783
Ahuja R K, Orlin J B. Combinatorial algorithms for inverse network flow problems. Networks, 2002, 40: 181–187
Burton D, Toint Ph L. On an instance of the inverse shortest path problem. Math Program, 1992, 53: 45–61
Burton D, Toint Ph L. On the use of an inverse shortest path algorithm for recovering linearly correlated costs. Math Program, 1994, 63: 1–22
Cao Y, Guan X. A class of constrained inverse bottleneck optimization problems under weighted hamming distance. In: Proceedings of the Second Joint Conference on Computational Sciences and Optimization, vol. 2. Washington: IEEE Computer Society, 2009, 859–863
Deaconu A. The inverse maximum flow problem considering l ∞-norm. RAIRO Oper Res, 2008, 42: 401–414
Fleischer L, Tardos E. Efficient continues-time dynamic network flow algorithms. Oper Res Lett, 1998, 23: 71–80
Ford L R, Fulkerson D R. Constructing maximal dynamic flows from static flows. Oper Res, 1958, 6: 419–433
Ford L R, Fulkerson D R. Flow in Networks. New Jersey: Princeton University Press, 1962
Gentry S, Saligrama V, Feron E. Dynamic Inverse Optimization. In: Proceedings of American Control Conference, vol. 6. Washington: IEEE Computer Society, 2001, 4722–4727
Guan X, Zhang J. Inverse bottleneck optimization problems under weighted hamming distance. In: Lecture Notes in Computer Science, vol. 4041. Berline: Springer-Verlag, 2006, 220–230
Hoppe B. Efficient dynamic network flow algorithms. PhD Thesis. Cornell University, 1995
Kostrzewski S. Inverse kinematics optimization and collision avoidance for KineMedic project. In: 31rd Summer European University on Surgical Robotics, 2007, Mont Pellier, France
Liu L, Zhang J. Inverse maximum flow problems under the weighted Hamming distance. J Comb Optim, 2006, 12: 395–408
Siemineski K. Direct solution of the inverse optimization problem of load sharing between muscles. J Biomechanics, 2006, 39: S45
Swamidas V J, Sherly S, Umesh M S, et al. Dosimetric comparison of inverse optimization with geometric optimization with graphical optimization for HDR prostate implants. J Med Phys, 2006, 31: 89–94
Tarantola A. Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation. Amsterdam: Elsevier, 1987
Yang C, Zhang J, Ma Z. Inverse maximum flow and minimum cut problems. Optimization, 1994, 40: 147–170
Zhang H. A solution to static inverse optimization problems with quadratic constraints by learning of neural networks. Trans Inst Elect Eng Japan, 2003, 123: 1901–1907
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bagherian, M. The inverse maximum dynamic flow problem. Sci. China Math. 53, 2709–2717 (2010). https://doi.org/10.1007/s11425-010-3129-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-3129-1