Polynomial cases of the tarification problem

We consider the problem of determining a set of optimal tariffs for an agent in a network, who owns a subset of the arcs of the network, and who wishes to maximize his revenues on this subset from a set of clients that make use of the network.The general variant of this problem is NP-hard, already with a single client. This paper introduces several new polynomially solvable special cases. An important case is the following.For multiple clients, if the number of tariff arcs is bounded from above, we can solve the problem by a polynomial number of linear programs (each of which is of polynomial size). Furthermore, we show that the parametric tarification problem and the single arc fixed charge tarification problem can be solved in polynomial time.Economics ;

Pricing Network Edges to Cross a River.

We consider a Stackelberg pricing problem in directed networks:Tariffs (prices) have to be defined by an operator, the leader, for a subset of the arcs. Clients, the followers, choose paths to route their demand through the network selfishly and independently of each other, on the basis of minimal total cost. The problem is to find tariffs such as to maximize the operator''s revenue. We consider the case where each client takes at most one tariff arc to route the demand.The problem, which we refer to as the river tarification problem, is a special case of problems studied previously in the literature.We prove that the problem is strongly NP-hard.Moreover, we show that the polynomially solvable case of uniform tarification yields an m--approximation algorithm, and this is tight. We suggest a new type of analysis that allows to improve the result to \bigO{\log m}, whenever the input data is polynomially bounded. We furthermore derive an \bigO{m^{1-\varepsilon}}--inapproximability result for problems where the operator must serve all clients, and we discuss some polynomial special cases. Finally, a computational study with instances from France Telecom suggests that uniform pricing performs better in practice than theory would suggest.operations research and management science;

Tariff optimization in Networks

We consider the problem of determining a set of optimal tariffs for an agent in the network, who owns a subset of all the arcs, and who receives revenue by setting the tariffs on the arc he owns. Multiple rational clients are active in the network, who route their demands on the cheapest paths from source to destination. The cost of a path is determined by fixed costs and tariffs on the arcs of the path.We introduce a remodeling of the network, using shortest paths. We develop three algorithms, a path oriented mixed integer program and a known arc oriented mixed integer program. Combined with reduction methods this remodeling enables us to solve the problem to optimality, for quite large instances. We provide computational results for the methods developped and compare them with the results of the arc oriented mixed integer programming formulation of the problem, applied to the original network.Economics ;

Linear Tarification in Multi-Commodity Telecommunications Networks

We consider the problem of determining a set ofoptimal tariffs for a revenue maximizing operator, on a subset ofall arcs of a telecommunications network. We suppose multiplerational clients are active on the network who route their demandson the cheapest paths from source to destination, where the costof a path is determined by all costs and tariffs on the arcs ofthe path.The complexity of the problem is studied first. Second, we proposea remodeling of the network, combined with model specific graphreduction methods. This new model is used in a branch and boundalgorithm to solve the problem to optimality. Finally, we providecomputational results which show the efficiency of our method fora number of real-life instances.economic systems ;

Pricing Bridges to Cross a River

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 + ln D)–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