Stackelberg Network Pricing is Hard to Approximate

TL;DR

This paper proves that the Stackelberg Network Pricing problem is not only NP-hard but also APX-hard, indicating it is difficult to approximate within any factor unless P=NP.

Contribution

The paper establishes the APX-hardness of the Stackelberg Network Pricing problem, strengthening previous NP-hardness and approximation results.

Findings

The problem is NP-hard.

The problem is APX-hard.

Approximation within any factor is unlikely.

Abstract

In the Stackelberg Network Pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte (Networks, Vol. 46(1), 57-67, 2005) proved that this problem is NP-hard, and gave an O(log m)-approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APX-hard.