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Competing for the most profitable tour: The orienteering interdiction game

Eduardo Álvarez-Miranda, Markus Sinnl, Kübra Tanınmış

TL;DR

The paper introduces the Orienteering Interdiction Game (OIG), a zero-sum bilevel extension of the orienteering problem where a leader interdicts nodes to minimize the follower’s best possible prize under a tour budget. It develops a single-level interdiction-cut reformulation (OIGS) and solves it with a comprehensive branch-and-cut algorithm, augmented by cut pools, solution pools, follower heuristics, and preprocessing; an exact solver is complemented by a genetic algorithm for fast high-quality solutions. Computational experiments on TSPLIB-based instances demonstrate that the enhanced B&C approach substantially reduces solving time, while the GA yields near-optimal solutions in substantially shorter timeframes. The work provides a versatile framework for competitive routing interdiction, with implications for security, canvassing, and patrol optimization, and points to future extensions to other orienteering variants and topological constraints.

Abstract

The orienteering problem is a well-studied and fundamental problem in transportation science. In the problem, we are given a graph with prizes on the nodes and lengths on the edges, together with a budget on the overall tour length. The goal is to find a tour that respects the length budget and maximizes the collected prizes. In this work, we introduce the orienteering interdiction game, in which a competitor (the leader) tries to minimize the total prize that the follower can collect within a feasible tour. To this end, the leader interdicts some of the nodes so that the follower cannot collect their prizes. The resulting interdiction game is formulated as a bilevel optimization problem, and a single-level reformulation is obtained based on interdiction cuts. A branch-and-cut algorithm with several enhancements, including the use of a solution pool, a cut pool and a heuristic method for the follower's problem, is proposed. In addition to this exact approach, a genetic algorithm is developed to obtain high-quality solutions in a short computing time. In a computational study based on instances from the literature for the orienteering problem, the usefulness of the proposed algorithmic components is assessed, and the branch-and-cut and genetic algorithms are compared in terms of solution time and quality.

Competing for the most profitable tour: The orienteering interdiction game

TL;DR

The paper introduces the Orienteering Interdiction Game (OIG), a zero-sum bilevel extension of the orienteering problem where a leader interdicts nodes to minimize the follower’s best possible prize under a tour budget. It develops a single-level interdiction-cut reformulation (OIGS) and solves it with a comprehensive branch-and-cut algorithm, augmented by cut pools, solution pools, follower heuristics, and preprocessing; an exact solver is complemented by a genetic algorithm for fast high-quality solutions. Computational experiments on TSPLIB-based instances demonstrate that the enhanced B&C approach substantially reduces solving time, while the GA yields near-optimal solutions in substantially shorter timeframes. The work provides a versatile framework for competitive routing interdiction, with implications for security, canvassing, and patrol optimization, and points to future extensions to other orienteering variants and topological constraints.

Abstract

The orienteering problem is a well-studied and fundamental problem in transportation science. In the problem, we are given a graph with prizes on the nodes and lengths on the edges, together with a budget on the overall tour length. The goal is to find a tour that respects the length budget and maximizes the collected prizes. In this work, we introduce the orienteering interdiction game, in which a competitor (the leader) tries to minimize the total prize that the follower can collect within a feasible tour. To this end, the leader interdicts some of the nodes so that the follower cannot collect their prizes. The resulting interdiction game is formulated as a bilevel optimization problem, and a single-level reformulation is obtained based on interdiction cuts. A branch-and-cut algorithm with several enhancements, including the use of a solution pool, a cut pool and a heuristic method for the follower's problem, is proposed. In addition to this exact approach, a genetic algorithm is developed to obtain high-quality solutions in a short computing time. In a computational study based on instances from the literature for the orienteering problem, the usefulness of the proposed algorithmic components is assessed, and the branch-and-cut and genetic algorithms are compared in terms of solution time and quality.
Paper Structure (24 sections, 1 theorem, 8 equations, 2 figures, 3 tables, 3 algorithms)

This paper contains 24 sections, 1 theorem, 8 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction and problem definition
  2. Contribution and outline
  3. Literature review
  4. An interdiction-cut-based exact solution method for the OIG
  5. A single-level reformulation of the OIG
  6. A branch-and-cut algorithm
  7. Separation of interdiction cuts
  8. Separation of follower cuts
  9. Generalized subtour elimination constraints.
  10. Logical constraints.
  11. Cycle cover inequalities.
  12. Enhancement strategies
  13. Separation problem objective lower bound (lower cutoff).
  14. Cut pool.
  15. Solution pool.
  16. ...and 9 more sections

Key Result

Proposition 1

The formulation (OIGS) models the OIG.

Figures (2)

  • Figure 1: The bayg29 instance from TSPLIB95 library, with a given depot node (orange) and considering unit prizes and unit interdiction costs. The interdicted nodes (green) and the resulting follower tour are shown under different leader budget $Q_\ell$.
  • Figure 2: Cumulative distribution of running times of all instances under different B&C settings.

Theorems & Definitions (3)

  • Definition 1: Orienteering interdiction game (OIG)
  • Proposition 1
  • proof