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

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 comple...

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Vydané v:	Mathematical programming Ročník 208; číslo 1-2; s. 425 - 461
Hlavní autori:	Bui, Quang Minh, Carvalho, Margarida, Neto, José
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2024 Springer Springer Verlag
Predmet:	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Pricing Theoretical 90C46 90C35 90C25 Network pricing problem Strong bilevel feasibility Conjugate model Complexity asymmetry
ISSN:	0025-5610, 1436-4646
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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, whose numerical results illustrate its potential for effectively solving the NPP with a high number of commodities.
AbstractList	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.
Audience	Academic
Author	Bui, Quang Minh Neto, José Carvalho, Margarida
Author_xml	– sequence: 1 givenname: Quang Minh orcidid: 0000-0002-3001-7538 surname: Bui fullname: Bui, Quang Minh email: quang.minh.bui@umontreal.ca organization: CIRRELT and Département d‘informatique et de recherche opérationnelle, Université de Montréal – sequence: 2 givenname: Margarida surname: Carvalho fullname: Carvalho, Margarida organization: CIRRELT and Département d‘informatique et de recherche opérationnelle, Université de Montréal – sequence: 3 givenname: José surname: Neto fullname: Neto, José organization: Télécom SudParis, Institut Polytechnique de Paris
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SubjectTerms	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Pricing Theoretical
Title	Asymmetry in the complexity of the multi-commodity network pricing problem
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