Pricing Bridges to Cross a River

WOAO Vol. 7, 2004

15 Pages Posted: 23 Jan 2006

See all articles by Alexander Grigoriev

Alexander Grigoriev

Maastricht University

Stan van Hoesel

Maastricht School of Business and Economics

Anton F. van der Kraaij

Maastricht University

Frits Spieksma

Catholic University of Leuven (KUL)

Marc Uetz

Maastricht University - Department of Economics

Mustapha Bouhtou

France Telecom R&D

Abstract

We consider a Stackelberg pricing problem in directed, uncapacitated networks. Tariffs have to be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs are assumed to be given. There are n clients, the followers, that route their demand independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by problems in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX-hard. Moreover, we show that uniform pricing yields both an m-approximation, and a (1 + lnD)-approximation. Here, D is upper bounded by the total demand of all clients. We furthermore discuss some polynomially solvable special cases, and present a short computational study with instances from France Télécom. In addition, we consider the problem under the additional restriction that the operator must serve all clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless NP = ZPP, and we prove the existence of an n-approximation algorithm.

Suggested Citation

Grigoriev, Alexander and van Hoesel, Stan and van der Kraaij, Anton F. and Spieksma, Frits and Uetz, Marc and Bouhtou, Mustapha, Pricing Bridges to Cross a River. Available at SSRN: https://ssrn.com/abstract=876587

Alexander Grigoriev (Contact Author)

Maastricht University ( email )

P.O. Box 616
Maastricht, 6200MD
Netherlands

Stan Van Hoesel

Maastricht School of Business and Economics ( email )

P.O. Box 616
Maastricht, 6200 MD
Netherlands

Anton F. Van der Kraaij

Maastricht University ( email )

P.O. Box 616
Maastricht, 6200MD
Netherlands

Frits Spieksma

Catholic University of Leuven (KUL) ( email )

Leuven, B-3000
Belgium

Marc Uetz

Maastricht University - Department of Economics ( email )

P.O. Box 616
Maastricht, 6200 MD
Netherlands

Mustapha Bouhtou

France Telecom R&D ( email )

39-40 rue de General Leclerc
Issy-Les-Moulineaux, F-92131
France

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