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Multi-Agent Route Planning as a QUBO Problem

Renáta Rusnáková, Martin Chovanec, Juraj Gazda

TL;DR

This work formalizes Multi-Agent Route Planning (MaRP) as selecting predefined vehicle routes to maximize unique network coverage while penalizing overlaps, and proves NP-hardness via a Weighted Set Packing reduction. It develops a QUBO formulation with coefficients that encode $u_i$ (unique coverage) and $c_{ij}$ (pairwise overlap), and distinguishes soft ($\lambda_{soft}$) and hard ($\lambda_{hard}$) penalty regimes to explore the coverage–overlap trade-off. A reproducible pipeline using OpenStreetMap networks and Valhalla routes demonstrates large-scale evaluation on Barcelona instances (up to $10^4$ vehicles) with solvers including Gurobi, simulated annealing, and D-Wave hybrid annealing; results show a pronounced coverage–overlap knee and that Pareto-optimal solutions largely occur under hard penalties, with hybrid quantum and classical solvers delivering essentially identical objective values at scale. The approach supports fleet sizing and scenario analysis as a practical planning tool, while acknowledging simplifications such as fixed routes and lack of temporal or capacity constraints, pointing to avenues for richer, time-aware extensions.

Abstract

Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that the spatial coverage of a road network is increased while redundant overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode unique coverage rewards and pairwise overlap penalties. A single penalty parameter controls the coverage-overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with an exact mixed-integer solver (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10 000 vehicles reveal a clear coverage-overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve essentially identical objective values with only minor differences in runtime as problem size grows.

Multi-Agent Route Planning as a QUBO Problem

TL;DR

This work formalizes Multi-Agent Route Planning (MaRP) as selecting predefined vehicle routes to maximize unique network coverage while penalizing overlaps, and proves NP-hardness via a Weighted Set Packing reduction. It develops a QUBO formulation with coefficients that encode (unique coverage) and (pairwise overlap), and distinguishes soft () and hard () penalty regimes to explore the coverage–overlap trade-off. A reproducible pipeline using OpenStreetMap networks and Valhalla routes demonstrates large-scale evaluation on Barcelona instances (up to vehicles) with solvers including Gurobi, simulated annealing, and D-Wave hybrid annealing; results show a pronounced coverage–overlap knee and that Pareto-optimal solutions largely occur under hard penalties, with hybrid quantum and classical solvers delivering essentially identical objective values at scale. The approach supports fleet sizing and scenario analysis as a practical planning tool, while acknowledging simplifications such as fixed routes and lack of temporal or capacity constraints, pointing to avenues for richer, time-aware extensions.

Abstract

Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that the spatial coverage of a road network is increased while redundant overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode unique coverage rewards and pairwise overlap penalties. A single penalty parameter controls the coverage-overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with an exact mixed-integer solver (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10 000 vehicles reveal a clear coverage-overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve essentially identical objective values with only minor differences in runtime as problem size grows.
Paper Structure (25 sections, 1 theorem, 16 equations)

This paper contains 25 sections, 1 theorem, 16 equations.

Table of Contents

  1. INTRODUCTION
  2. Problem Definition
  3. NP-Hardness via Reduction from Weighted Set Packing
  4. QUBO Formulation of MaRP
  5. Encoding Coverage and Overlap
  6. Penalty Weights: $\lambda_{\text{soft}}$ vs. $\lambda_{\text{hard}}$
  7. Simulation Setup and Metrics
  8. Instance Generation and Implementation
  9. Solvers
  10. Post-optimization Evaluation Metrics
  11. Average node overlap.
  12. Top 10% overlap share.
  13. Overlap graph density and average degree.
  14. Normalized Shannon entropy Shannon1948.
  15. Herfindahl--Hirschman Index (HHI) Brezina2016.
  16. ...and 10 more sections

Key Result

Theorem 3.1

The MaRP optimization problem in obj is NP-hard.

Theorems & Definitions (4)

  • Definition 3.1: Weighted Set Packing
  • Theorem 3.1
  • proof
  • Remark 3.1