Asymmetry in the complexity of the multi-commodity network pricing problem

• 成果类型: Article
• 署名作者: Bui, Quang Minh; Carvalho, Margarida; Neto, Jose
• 署名单位: Universite de Montreal; Universite de Montreal; IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom SudParis
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-02043-2
• 发表日期: 2024
• 页码: 425-461
• 关键词: bilevel model; optimization; formulations; algorithm

摘要

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, whose numerical results illustrate its potential for effectively solving the NPP with a high number of commodities.