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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Bibliographic Details
Published in:Mathematical programming Vol. 208; no. 1-2; pp. 425 - 461
Main Authors: Bui, Quang Minh, Carvalho, Margarida, Neto, José
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2024
Springer
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-02043-2