Bottleneck capacity expansion problems with general budget constraints

Rainer E. Burkard; Bettina Klinz; Jianzhong Zhang

RAIRO - Operations Research - Recherche Opérationnelle (2001)

  • Volume: 35, Issue: 1, page 1-20
  • ISSN: 0399-0559

Abstract

top
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 e E . Moreover, we are given monotone increasing cost functions f e for increasing the capacity of the elements e E 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 e F 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].

How to cite

top

Burkard, Rainer E., Klinz, Bettina, and Zhang, Jianzhong. "Bottleneck capacity expansion problems with general budget constraints." RAIRO - Operations Research - Recherche Opérationnelle 35.1 (2001): 1-20. <http://eudml.org/doc/105235>.

@article{Burkard2001,
abstract = {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 $\{\mathcal \{F\}\}$ of feasible subsets of $E$ and a nonnegative real capacity $\widehat\{c\}_\{e\}$ for all $e\in E$. Moreover, we are given monotone increasing cost functions $f_\{e\}$ for increasing the capacity of the elements $e\in E$ as well as a budget $B$. The task is to determine new capacities $c_\{e\}\ge \widehat\{c\}_\{e\}$ such that the objective function given by $\max _\{F\in \{\mathcal \{F\}\}\}\min _\{e\in F\} c_\{e\}$ 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].},
author = {Burkard, Rainer E., Klinz, Bettina, Zhang, Jianzhong},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {capacity expansion; bottleneck problem; strongly polynomial algorithm; algebraic optimization; algebraic optimization. },
language = {eng},
number = {1},
pages = {1-20},
publisher = {EDP-Sciences},
title = {Bottleneck capacity expansion problems with general budget constraints},
url = {http://eudml.org/doc/105235},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Burkard, Rainer E.
AU - Klinz, Bettina
AU - Zhang, Jianzhong
TI - Bottleneck capacity expansion problems with general budget constraints
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 1
SP - 1
EP - 20
AB - 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 ${\mathcal {F}}$ of feasible subsets of $E$ and a nonnegative real capacity $\widehat{c}_{e}$ for all $e\in E$. Moreover, we are given monotone increasing cost functions $f_{e}$ for increasing the capacity of the elements $e\in E$ as well as a budget $B$. The task is to determine new capacities $c_{e}\ge \widehat{c}_{e}$ such that the objective function given by $\max _{F\in {\mathcal {F}}}\min _{e\in F} c_{e}$ 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].
LA - eng
KW - capacity expansion; bottleneck problem; strongly polynomial algorithm; algebraic optimization; algebraic optimization.
UR - http://eudml.org/doc/105235
ER -

References

top
  1. [1] R.K. Ahuja and J.B. Orlin, A capacity scaling algorithm for the constrained maximum flow problem. Networks 25 (1995) 89-98. Zbl0821.90041
  2. [2] R.E. Burkard, K. Dlaska and B. Klinz, The quickest flow problem. Z. Oper. Res. (ZOR) 37 (1993) 31-58. Zbl0780.90031MR1213677
  3. [3] R.E. Burkard, W. Hahn and U. Zimmermann, An algebraic approach to assignment problems. Math. Programming 12 (1977) 318-327. Zbl0361.90047MR456526
  4. [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. Zbl0483.90086MR663201
  5. [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. Zbl0913.68144MR1632632
  6. [6] G.N. Frederickson and R. Solis-Oba, Increasing the weight of minimum spanning trees. J. Algorithms 33 (1999) 244-266. Zbl0956.68113MR1718739
  7. [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. Zbl1321.05038MR1447714
  8. [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. Zbl0366.90115MR489860
  9. [9] A. Jüttner, On budgeted optimization problems. Private Communication (2000). Zbl1136.90036
  10. [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. Zbl0916.90261MR1667012
  11. [11] N. Megiddo, Combinatorial optimization with rational objective functions. Math. Oper. Res. 4 (1979) 414-424. Zbl0425.90076MR549127
  12. [12] N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30 (1983) 852-865. Zbl0627.68034MR819134
  13. [13] C. Phillips, The network inhibition problem, in Proc. of the 25 th Annual Symposium on the Theory of Computing (1993) 776-785. Zbl1310.90018
  14. [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. Zbl0968.90515MR1358852
  15. [15] J. Zhang, C. Yang and Y. Lin, A class of bottleneck expansion problems. Comput. Oper. Res. 28 (2001) 505-519. Zbl0991.90113
  16. [16] U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures. North-Holland, Amsterdam, Ann. Discrete Math. 10 (1981). Zbl0466.90045MR609751

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.