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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Vydáno v:Mathematical programming Ročník 208; číslo 1-2; s. 425 - 461
Hlavní autoři: Bui, Quang Minh, Carvalho, Margarida, Neto, José
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2024
Springer
Springer Verlag
Témata:
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
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  organization: CIRRELT and Département d‘informatique et de recherche opérationnelle, Université de Montréal
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  surname: Carvalho
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  givenname: José
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  organization: Télécom SudParis, Institut Polytechnique de Paris
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ContentType Journal Article
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COPYRIGHT 2024 Springer
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Issue 1-2
Keywords 90C46
90C35
90C25
Network pricing problem
Strong bilevel feasibility
Conjugate model
Complexity asymmetry
Language English
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Snippet 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...
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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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