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Asymmetry in the Complexity of the Multi-Commodity Network Pricing Problem

Quang Minh Bui, Margarida Carvalho, José Neto

TL;DR

This paper 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, and characterize this asymmetry in the complexity with a novel property named strong bilevel feasibility.

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, accommodated with numerical results to demonstrate its effectiveness in solving the NPP with a high number of commodities.

Asymmetry in the Complexity of the Multi-Commodity Network Pricing Problem

TL;DR

This paper 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, and characterize this asymmetry in the complexity with a novel property named strong bilevel feasibility.

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, accommodated with numerical results to demonstrate its effectiveness in solving the NPP with a high number of commodities.
Paper Structure (19 sections, 18 theorems, 38 equations, 12 figures, 3 tables, 2 algorithms)

This paper contains 19 sections, 18 theorems, 38 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions and Paper Organization
  4. Asymmetry in the Complexity
  5. Single-Commodity Case
  6. Single-Tolled-Arc Case
  7. Reaction Plot
  8. Conjugate Model
  9. Strong Bilevel Feasibility
  10. Bilevel Feasibility
  11. Action-Reaction Duality
  12. Strong Bilevel Feasibility
  13. Composition
  14. Bilevel Feasibility
  15. Strong Bilevel Feasibility
  16. ...and 4 more sections

Key Result

Theorem 1

If the number of tolled arcs $|\mathcal{A}_1|$ is fixed, then the multi-commodity NPP can be solved in polynomial time with respect to the number of commodities $|\mathcal{K}|$, specifically by solving $(|\mathcal{K}| + 1)^{|\mathcal{A}_1|}$ linear programs, each with size polynomial in $|\mathcal{A

Figures (12)

  • Figure 1: Graph for \ref{['ex:single-comm']}.
  • Figure 2: Illustrations for \ref{['ex:single-toll']}.
  • Figure 3: Examples of reaction plots.
  • Figure 4: Composition of the reaction plot in \ref{['ex:multi-comm']}.
  • Figure 5: $H_=(g(w), w)$ is the highest hyperplane given $w$.
  • ...and 7 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Proposition 2
  • proof
  • proof : Proof of \ref{['thm:multi-comm-poly']}
  • Example 4
  • Definition 3
  • Lemma 4: Bui et al. bui2022
  • ...and 38 more