2006 | OriginalPaper | Buchkapitel
How to Sell a Graph: Guidelines for Graph Retailers
verfasst von : Alexander Grigoriev, Joyce van Loon, René Sitters, Marc Uetz
Erschienen in: Graph-Theoretic Concepts in Computer Science
Verlag: Springer Berlin Heidelberg
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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
.