Joyce van Loon

How to sell a graph: guidelines for graph retailers

We consider a profit maximization problem where we are asked to price a set of m items that are to be assigned to a set of n customers. The items can be represented as the edges of an undirected (multi)graph G, where an edge multiplicity larger than one corresponds to multiple copies of the same item. Each customer is interested in purchasing a bundle of edges of G, and we assume that each bundle forms a simple path in G. Each customer has a known budget for her respective bundle, and is interested only in that particular bundle. The goal is to determine item prices and a feasible assignment of items to customers in order to maximize the total profit. When the underlying graph G is a path, we derive a fully polynomial time approximation scheme, complementing a recent NP-hardness result. If the underlying graph is a tree, and edge multiplicities are one, we show that the problem is polynomially solvable, contrasting its APX-hardness for the case of unlimited availability of items. However, if the underlying graph is a grid, and edge multiplicities are one, we show that it is even NP-complete to approximate the maximum profit to within a factor n1−−e.

Price strategy implementation

Consider a situation in which a company sells several different items to a set of customers. However, the company is not satisfied with the current pricing strategy and wishes to implement new prices for its items. Implementing these new prices in one single step might not be desirable, for example, because of the change in contract prices for the customers. Therefore, the company changes the prices gradually, such that the prices charged to a subset of the customers, the target market, do not differ too much from one period to the next. We propose a polynomial time algorithm to implement the new prices in the minimum number of time periods needed, given that the prices charged to the customers in the target market increase by at most a factor 1 + ? , for a given ? 0 . Furthermore, we address the problem of maximizing the overall revenue during the price implementation over a given time horizon. For this problem, we describe a dynamic program for the case of integer price vectors, and a local search algorithm for arbitrary prices. Also, we present a mixed integer programming formulation for this problem and apply our algorithms in a practical study.

On the complexity of a bundle pricing problem

We consider the problem of pricing items in order to maximize the revenue obtainable from a set of single minded customers. We relate the tractability of the problem to structural properties of customers’ valuations: the problem admits an efficient approximation algorithm, parameterized along the inhomogeneity of the valuations.

Algorithms for Optimal Price Regulations

Since summer 2007, mobile phone users in the European Union (EU) are protected by a ceiling on the roaming tariff when calling or receiving a call abroad. We analyze the effects of this price regulative policy, and compare it to alternative implementations of price regulations. The problem is a three-level mathematical program: The EU determines the price regulative policy, the telephone operator sets profit-maximizing prices, and customers choose to accept or decline the operators offer. The first part of this paper contains a polynomial time algorithm to solve such a three-level program. The crucial idea is to partition the polyhedron of feasible price regulative parameters into a polynomial number of smaller polyhedra such that a certain primitive decision problem can be written as an LP on each of those. Then the problem can be solved by a combination of enumeration and linear programming. In the second part, we analyze more specifically an instance of this problem, namely the price regulation problem that the EU encounters. Using customer-data from a large telephone operator, we compare different price regulative policies with respect to their social welfare. On the basis of the specific social welfare function, we observe that other price regulative policies or different ceilings can improve the total social welfare.