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 n 1−ε .

Abstract. We consider a revenue maximization problem where we are selling a set of items, each available in a certain quantity, to a set of bidders. Each bidder is interested in one or several bundles of items. We assume the bidders' valuations for each of these bundles to be known. Whenever bundle prices are determined by the sum of single item prices, this algorithmic problem was recently shown to be inapproximable to within a semi-logarithmic factor. We consider two scenarios for determining bundle prices that allow to break this inapproximability barrier. Both scenarios are motivated by problems where items are different, yet comparable. First, we consider classical single item prices with an additional monotonicity constraint, enforcing that larger bundles are at least as expensive as smaller ones. We show that the problem remains strongly NP-hard, and we derive a PTAS. Second, motivated by real-life cases, we introduce the notion of affine price functions, and derive fixed-parameter polynomial time algorithms.

We address the algorithmic complexity of a profit maximization problem in capacitated, undirected networks. We are asked to price a set of m capacitated network links to serve a set of n potential customers. Each customer is interested in purchasing a network connection that is specified by a simple path in the network and has a maximum budget that we assume to be known to the seller. The goal is to decide which customers to serve, and to determine prices for all network links in order to maximize the total profit. We address this pricing problem in different network topologies. More specifically, we derive several results on the algorithmic complexity of this profit maximization problem, given that the network is either a path, a cycle, a tree, or a grid. Our results include approximation algorithms as well as inapproximability results.

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