Abstract. We consider the problem of pricing items so as to maximize the profit made from selling these items. An instance is given by a set E of n items and a set of m clients, where each client is specified by one subset of E (the bundle of items she/he wants to buy), and a budget (valuation), which is the maximum price she/he is willing to pay for that subset. We restrict our attention to the model where the subsets can be arranged such that they form intervals of a line graph. Assuming an unlimited supply of any item, this problem is known as the highway problem and so far only an O(log n)-approximation algorithm is known. We show that a PTAS is likely to exist by presenting a quasi-polynomial time approximation scheme. We also combine our ideas with a recently developed quasi-PTAS for the unsplittable flow problem on line graphs to extend this approximation scheme to the limited supply version of the pricing problem.
Finally, Elbassioni et al. REF present a quasi-polynomial time approximation scheme for both the capacitated and uncapacitated version of the problem, thereby suggesting that a PTAS is likely to exist.
A quasi-PTAS for profit-maximizing pricing on line graphs
In Proc. 15th Ann. European Symp. on Algorithms (ESA’07), Lecture Notes in Computer Science
Computer Science