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A Catalog of Formulations for the Network Pricing Problem

Quang Minh Bui, Bernard Gendron, Margarida Carvalho

TL;DR

This work addresses the network pricing problem (NPP), a bilevel optimization where a leader sets tolls to maximize revenue given rational followers selecting shortest paths. It develops a unified framework that links 12 single-level reformulations by mixing primal/dual representations with strong duality or complementary slackness, and introduces a novel path enumeration scheme and path-based preprocessing. A hybrid multi-commodity framework is proposed to adapt reformulations per commodity, balancing the cost of path enumeration with solving the resulting MILPs. Empirical results show that path-based preprocessing markedly reduces problem size and often outperforms SPGM, while the STD formulation remains highly competitive; the hybrid approach provides robust performance across instance classes. These contributions offer a practical toolkit for efficiently solving NPP-like bilevel problems and guidance for selecting reformulations and preprocessing in bilevel contexts.

Abstract

We study the network pricing problem where the leader maximizes their revenue by determining the optimal amounts of tolls to charge on a set of arcs, under the assumption that the followers will react rationally and choose the shortest paths to travel. Many distinct single-level reformulations to this bilevel optimization program have been proposed, however, their relationship has not been established. In this paper, we aim to build a connection between those reformulations and explore the combination of the path representation with various modeling options, allowing us to generate 12 different reformulations of the problem. Moreover, we propose a new path enumeration scheme, path-based preprocessing, and hybrid framework to further improve performance and robustness when solving the final model. We provide numerical results, comparing all the derived reformulations and confirming the efficiency of the novel dimensionality reduction procedures.

A Catalog of Formulations for the Network Pricing Problem

TL;DR

This work addresses the network pricing problem (NPP), a bilevel optimization where a leader sets tolls to maximize revenue given rational followers selecting shortest paths. It develops a unified framework that links 12 single-level reformulations by mixing primal/dual representations with strong duality or complementary slackness, and introduces a novel path enumeration scheme and path-based preprocessing. A hybrid multi-commodity framework is proposed to adapt reformulations per commodity, balancing the cost of path enumeration with solving the resulting MILPs. Empirical results show that path-based preprocessing markedly reduces problem size and often outperforms SPGM, while the STD formulation remains highly competitive; the hybrid approach provides robust performance across instance classes. These contributions offer a practical toolkit for efficiently solving NPP-like bilevel problems and guidance for selecting reformulations and preprocessing in bilevel contexts.

Abstract

We study the network pricing problem where the leader maximizes their revenue by determining the optimal amounts of tolls to charge on a set of arcs, under the assumption that the followers will react rationally and choose the shortest paths to travel. Many distinct single-level reformulations to this bilevel optimization program have been proposed, however, their relationship has not been established. In this paper, we aim to build a connection between those reformulations and explore the combination of the path representation with various modeling options, allowing us to generate 12 different reformulations of the problem. Moreover, we propose a new path enumeration scheme, path-based preprocessing, and hybrid framework to further improve performance and robustness when solving the final model. We provide numerical results, comparing all the derived reformulations and confirming the efficiency of the novel dimensionality reduction procedures.

Paper Structure

This paper contains 19 sections, 7 theorems, 33 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Contribution and Paper Organization
  4. The Network Pricing Problem
  5. Problem Formulation
  6. Generalized Single-level Reformulations
  7. Linearization
  8. Linearization of Strong Duality Formulations
  9. Linearization of Complementary Slackness Formulations
  10. Path Enumeration and Preprocessing
  11. Path Enumeration
  12. Path-based Preprocessing
  13. Hybrid Model for Multi-Commodity Problems
  14. Experiments
  15. Methodology
  16. ...and 4 more sections

Key Result

Lemma 2

Consider any commodity $k \in K$. Let $p$ and $q$ be two different paths in $P^k$. If for all $a \in A_1$, $\delta_a^p \leq \delta_a^q$ and then $q$ is not bilevel feasible, i.e., $q$ cannot be the optimal path for any value of $T$.

Figures (7)

  • Figure 1: Graph with redundant paths (dashed arcs are tolled arcs).
  • Figure 2: Illustration of the three types of topology.
  • Figure 3: Cumulative distribution of the number of bilevel feasible paths.
  • Figure 4: Performance of notable formulations over $N$.
  • Figure 5: Reduction ratio of Path-based Preprocessing.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1
  • Lemma 2: Bouhtou et al. bouhtou2007
  • proof
  • Theorem 3
  • proof
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 5 more