Bottleneck capacity expansion problems with general budget constraints
RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 1, pp. 1-20.

This paper presents a unified approach for bottleneck capacity expansion problems. In the bottleneck capacity expansion problem, BCEP, we are given a finite ground set E, a family of feasible subsets of E and a nonnegative real capacity c ^ e for all eE. Moreover, we are given monotone increasing cost functions f e for increasing the capacity of the elements eE as well as a budget B. The task is to determine new capacities c e c ^ e such that the objective function given by max F min eF c e is maximized under the side constraint that the overall expansion cost does not exceed the budget B. We introduce an algebraic model for defining the overall expansion cost and for formulating the budget constraint. This models allows to capture various types of budget constraints in one general model. Moreover, we discuss solution approaches for the general bottleneck capacity expansion problem. For an important subclass of bottleneck capacity expansion problems we propose algorithms which perform a strongly polynomial number of steps. In this manner we generalize and improve a recent result of Zhang et al. [15].

Classification : 90C57,  90C31,  90C32
Mots clés : capacity expansion, bottleneck problem, strongly polynomial algorithm, algebraic optimization
@article{RO_2001__35_1_1_0,
     author = {Burkard, Rainer E. and Klinz, Bettina and Zhang, Jianzhong},
     title = {Bottleneck capacity expansion problems with general budget constraints},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1--20},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {1},
     year = {2001},
     zbl = {1078.90585},
     mrnumber = {1841811},
     language = {en},
     url = {http://www.numdam.org/item/RO_2001__35_1_1_0/}
}
Burkard, Rainer E.; Klinz, Bettina; Zhang, Jianzhong. Bottleneck capacity expansion problems with general budget constraints. RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 1, pp. 1-20. http://www.numdam.org/item/RO_2001__35_1_1_0/

[1] R.K. Ahuja and J.B. Orlin, A capacity scaling algorithm for the constrained maximum flow problem. Networks 25 (1995) 89-98. | Zbl 0821.90041

[2] R.E. Burkard, K. Dlaska and B. Klinz, The quickest flow problem. Z. Oper. Res. (ZOR) 37 (1993) 31-58. | MR 1213677 | Zbl 0780.90031

[3] R.E. Burkard, W. Hahn and U. Zimmermann, An algebraic approach to assignment problems. Math. Programming 12 (1977) 318-327. | MR 456526 | Zbl 0361.90047

[4] R.E. Burkard and U. Zimmermann, Combinatorial optimization in linearly ordered semimodules: A survey, in Modern Applied Mathematics, edited by B. Korte. North Holland, Amsterdam (1982) 392-436. | MR 663201 | Zbl 0483.90086

[5] 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. 203 (1998) 91-121. | MR 1632632 | Zbl 0913.68144

[6] G.N. Frederickson and R. Solis-Oba, Increasing the weight of minimum spanning trees. J. Algorithms 33 (1999) 244-266. | MR 1718739 | Zbl 0956.68113

[7] G.N. Frederickson and R. Solis-Oba, Algorithms for robustness in matroid optimization, in Proc. of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (1997) 659-668. | MR 1447714

[8] D.R. Fulkerson and G.C. Harding, Maximizing the minimum source-sink path subject to a budget constraint. Math. Programming 13 (1977) 116-118. | MR 489860 | Zbl 0366.90115

[9] A. Jüttner, On budgeted optimization problems. Private Communication (2000).

[10] S.O. Krumke, M.V. Marathe, H. Noltemeier, R. Ravi and S.S. Ravi, Approximation algorithms for certain network improvement problems. J. Combin. Optim. 2 (1998) 257-288. | MR 1667012 | Zbl 0916.90261

[11] N. Megiddo, Combinatorial optimization with rational objective functions. Math. Oper. Res. 4 (1979) 414-424. | MR 549127 | Zbl 0425.90076

[12] N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30 (1983) 852-865. | MR 819134 | Zbl 0627.68034

[13] C. Phillips, The network inhibition problem1993) 776-785.

[14] T. Radzik, Parametric flows, weighted means of cuts, and fractional combinatorial optimization, in Complexity in Numerical Optimization, edited by P.M. Pardalos. World Scientific Publ. (1993) 351-386. | MR 1358852 | Zbl 0968.90515

[15] J. Zhang, C. Yang and Y. Lin, A class of bottleneck expansion problems. Comput. Oper. Res. 28 (2001) 505-519. | Zbl 0991.90113

[16] U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures. North-Holland, Amsterdam, Ann. Discrete Math. 10 (1981). | MR 609751 | Zbl 0466.90045