Asymmetry in the Complexity of the Multi-Commodity Network Pricing Problem

TL;DR

This paper explores the complexity of the network pricing problem, revealing an asymmetry based on the number of tolled arcs and commodities, and introduces algorithms and properties to address this challenge.

Contribution

It characterizes the asymmetry in complexity related to tolled arcs and commodities, and proposes algorithms leveraging strong bilevel feasibility.

Findings

Polynomial-time solvability when tolled arcs are fixed

NP-hardness with even one commodity if tolled arcs are not fixed

Effective inequality generation algorithm demonstrated

Abstract

The network pricing problem (NPP) is a bilevel problem, where the leader optimizes its revenue by deciding on the prices of certain arcs in a graph, while expecting the followers (also known as the commodities) to choose a shortest path based on those prices. In this paper, we investigate the complexity of the NPP with respect to two parameters: the number of tolled arcs, and the number of commodities. We devise a simple algorithm showing that if the number of tolled arcs is fixed, then the problem can be solved in polynomial time with respect to the number of commodities. In contrast, even if there is only one commodity, once the number of tolled arcs is not fixed, the problem becomes NP-hard. We characterize this asymmetry in the complexity with a novel property named strong bilevel feasibility. Finally, we describe an algorithm to generate valid inequalities to the NPP based on this property, accommodated with numerical results to demonstrate its effectiveness in solving the NPP with a high number of commodities.