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Edge downgrades in the maximal covering location problem

Marta Baldomero-Naranjo, Jörg Kalcsics, Antonio M. Rodríguez-Chía

TL;DR

This paper introduces the Downgrading Maximal Covering Location Problem (D-MCLP), a novel bilevel framework where a defender locates $p$ facilities to maximize post-downgrade coverage while an attacker, constrained by budget $B$, downgrades edge lengths to erode initial coverage. The authors develop a mixed-integer linear bilevel formulation, a preprocessing step to prune the problem, and a matheuristic comprising an Alternating Location-Downgrading Search and a 1-1 Local Search to obtain high-quality solutions efficiently. They provide exact results on moderate instances, demonstrate the substantial benefits of preprocessing, and show that the matheuristic achieves near-optimal solutions in a fraction of the time, outperforming naive sequential approaches in managerial insight experiments. The work lays groundwork for robust network design under adversarial disruption and suggests directions for discrete downgrading, node-downgrading, and stronger exact methods.

Abstract

We tackle the downgrading maximal covering location problem within a network. In this problem, two actors with conflicting objectives are involved: (a) The location planner aims to determine the location of facilities to maximize the covered demand while anticipating that an attacker will attempt to reduce coverage by increasing the length of some edges (downgrade); (b) The attacker seeks to maximize the demand initially covered by the facilities but left uncovered after the downgrade. The attacker can increase the length of certain edges within a specified budget. We introduce a mixed-integer linear bilevel program to formulate the problem, followed by a preprocessing phase and a matheuristic algorithm designed to address it. Additionally, computational results are presented to illustrate the potential and limitations of the proposed algorithm.

Edge downgrades in the maximal covering location problem

TL;DR

This paper introduces the Downgrading Maximal Covering Location Problem (D-MCLP), a novel bilevel framework where a defender locates facilities to maximize post-downgrade coverage while an attacker, constrained by budget , downgrades edge lengths to erode initial coverage. The authors develop a mixed-integer linear bilevel formulation, a preprocessing step to prune the problem, and a matheuristic comprising an Alternating Location-Downgrading Search and a 1-1 Local Search to obtain high-quality solutions efficiently. They provide exact results on moderate instances, demonstrate the substantial benefits of preprocessing, and show that the matheuristic achieves near-optimal solutions in a fraction of the time, outperforming naive sequential approaches in managerial insight experiments. The work lays groundwork for robust network design under adversarial disruption and suggests directions for discrete downgrading, node-downgrading, and stronger exact methods.

Abstract

We tackle the downgrading maximal covering location problem within a network. In this problem, two actors with conflicting objectives are involved: (a) The location planner aims to determine the location of facilities to maximize the covered demand while anticipating that an attacker will attempt to reduce coverage by increasing the length of some edges (downgrade); (b) The attacker seeks to maximize the demand initially covered by the facilities but left uncovered after the downgrade. The attacker can increase the length of certain edges within a specified budget. We introduce a mixed-integer linear bilevel program to formulate the problem, followed by a preprocessing phase and a matheuristic algorithm designed to address it. Additionally, computational results are presented to illustrate the potential and limitations of the proposed algorithm.

Paper Structure

This paper contains 20 sections, 4 theorems, 19 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Overview
  4. Problem description and mathematical formulations
  5. Trilevel Problem
  6. Lower level problem
  7. Middle level problem
  8. Upper level problem
  9. Preprocessing
  10. Matheuristics
  11. Alternating Location-Downgrading Search
  12. 1-1 Local Search
  13. Computational Results
  14. Data
  15. Results
  16. ...and 5 more sections

Key Result

Lemma 1

Formulation constr:Level3:Dual2:ObjectiveFunction$-$constr:Level3:Dual2:VarDefinition2 has an optimal solution $(\pi,\mu)$ with $\pi_k^i \ge 0$ for $k \in V$, $\pi_t^i=0$, and $\mu_k=0$ whenever $\overline x_k=1$, $k \in V$.

Figures (6)

  • Figure 1: Graphical representation of the improvement.
  • Figure 2: Performance profile graph of #solved instances (out of 135) for $n=50$.
  • Figure 3: Comparison of time for $n=50$ by matheuristic variants.
  • Figure 4: Comparison of $\%G_{BS}$ in one hour for $n=50$ by matheuristic variants.
  • Figure 5: Comparison of Managerial Insights for $n=50$ of Optimal Out-In Variant.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Lemma 3