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A Greedy Quantum Route-Generation Algorithm

Jordan Makansi, David E. Bernal Neira

TL;DR

The paper tackles routing with time windows by formulating the Fleet Sizing Vehicle Routing Problem with Time Windows ($FSVRPTW$) as a QUBO and proposes a greedy quantum route-generation algorithm that leverages information from all QC samples. By representing active problem variables as a directed acyclic graph (DAG) and iteratively extracting feasible sub-paths, the method converges to a feasible solution and, with exact subproblem solving, to optimality. Empirical results on D-Wave hardware show the approach can achieve lower objective values than state-of-the-art annealing-based methods within the same time budgets, and it demonstrates robustness to quantum noise compared to simple sample-filtering approaches. The work highlights the potential of integrating QC samples with adaptive, graph-based routing constructs, and suggests avenues for tuning parameters and applying the framework to broader constrained optimization problems in logistics.

Abstract

Routing and scheduling problems with time windows have long been important optimization problems for logistics and planning. Many classical heuristics and exact methods exist for such problems. However, there are no satisfactory methods for generating routes using quantum computing (QC), for mainly two reasons: inequality constraints, and the trade-off of feasibility and solution quality. Inequality constraints are typically handled using slack variables; and feasible solutions are found by filtering samples. These challenges are amplified in the presence of noise inherent in QC. Here, we propose a greedy algorithm that generates routes by using information from all samples obtained from the quantum computer. By noticing the relationship between qubits in our formulation as a directed acyclic graph (DAG), we designed an algorithm that adaptively constructs a feasible solution. We prove its convergence to a feasible solution, and illustrate its efficacy by solving the Fleet Sizing Vehicle Routing Problem with Time Windows (FSVRPTW). Our computational results show that this method obtains a lower objective value than the current state-of-the-art annealing approaches, both classical and hybrid, for the same amount of time using D-Wave Hybrid Solvers. We also show its robustness to noise on D-Wave Advantage2 through computational results as compared to the filtering approach on DWaveSampler, even when the filtering approach is given a longer annealing time, and a larger sample size.

A Greedy Quantum Route-Generation Algorithm

TL;DR

The paper tackles routing with time windows by formulating the Fleet Sizing Vehicle Routing Problem with Time Windows () as a QUBO and proposes a greedy quantum route-generation algorithm that leverages information from all QC samples. By representing active problem variables as a directed acyclic graph (DAG) and iteratively extracting feasible sub-paths, the method converges to a feasible solution and, with exact subproblem solving, to optimality. Empirical results on D-Wave hardware show the approach can achieve lower objective values than state-of-the-art annealing-based methods within the same time budgets, and it demonstrates robustness to quantum noise compared to simple sample-filtering approaches. The work highlights the potential of integrating QC samples with adaptive, graph-based routing constructs, and suggests avenues for tuning parameters and applying the framework to broader constrained optimization problems in logistics.

Abstract

Routing and scheduling problems with time windows have long been important optimization problems for logistics and planning. Many classical heuristics and exact methods exist for such problems. However, there are no satisfactory methods for generating routes using quantum computing (QC), for mainly two reasons: inequality constraints, and the trade-off of feasibility and solution quality. Inequality constraints are typically handled using slack variables; and feasible solutions are found by filtering samples. These challenges are amplified in the presence of noise inherent in QC. Here, we propose a greedy algorithm that generates routes by using information from all samples obtained from the quantum computer. By noticing the relationship between qubits in our formulation as a directed acyclic graph (DAG), we designed an algorithm that adaptively constructs a feasible solution. We prove its convergence to a feasible solution, and illustrate its efficacy by solving the Fleet Sizing Vehicle Routing Problem with Time Windows (FSVRPTW). Our computational results show that this method obtains a lower objective value than the current state-of-the-art annealing approaches, both classical and hybrid, for the same amount of time using D-Wave Hybrid Solvers. We also show its robustness to noise on D-Wave Advantage2 through computational results as compared to the filtering approach on DWaveSampler, even when the filtering approach is given a longer annealing time, and a larger sample size.
Paper Structure (17 sections, 5 theorems, 4 equations, 5 figures, 6 tables, 3 algorithms)

This paper contains 17 sections, 5 theorems, 4 equations, 5 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Pre-processing
  5. Methods
  6. Greedy Quantum Route-Generation Algorithm
  7. DAG Representation
  8. Pruning
  9. Pruning Rules
  10. Analysis
  11. Computational Results
  12. Results - greedy+CQM, SAS, and CQM
  13. Results - greedy+SAS vs CQM
  14. Results - greedy+QPU vs CQM
  15. Results - greedy+QPU vs QPU
  16. ...and 2 more sections

Key Result

Lemma 1

In any iteration $l$, if a customer $i^P_k$ is in the interior of path $P \in S^l$, all outgoing and incoming variables to that customer have been pruned.

Figures (5)

  • Figure 1: Timetable and distance matrix for $N$=2 example. Initial set of variables $X^0$ represented as a DAG (denoted DAG($X^0$)), continued from example shown in the Table, labeled with possible one-body expectation values from step \ref{['exp_end']} in Alg. \ref{['alg:greedy']}. Variables selected in step \ref{['select']} in Alg. \ref{['alg:greedy']}, using $\theta = 0.5$, are marked with solid arrows. Paths found in step \ref{['find_paths']} in Alg. \ref{['alg:greedy']} are highlighted in blue.
  • Figure 2: Pruning example for $\theta$=0.9, and subsequent iteration of Alg. \ref{['alg:greedy']}
  • Figure 3: Average relative optimality gap for SAS, and CQM and greedy+CQM, averaged over 10 examples with standard deviation. The relative optimal value is computed as $\frac{C_{A}-C_{opt}}{C_{opt}}$ where $C_A$ is the objective value found by the annealing method, and $C_{opt}$ is the optimal value found by Gurobi. Statistics are computed only over feasible solutions. Data not shown for SAS, $N$=50 - no feasible solutions found.
  • Figure 4: Relative time difference between greedy+CQM and SAS /CQM averaged over ten examples. Rel. time difference is computed as $\frac{t_{A}-t_{greedy}}{t_{A}}$ where $t_A$ is the time of SAS/CQM, and $t_{greedy}$ is time taken of greedy+CQM. Statistics are computed only over feasible solutions. Data not shown for SAS, $N$=50, because no feasible solutions found.
  • Figure 5: Average relative optimality gap comparing greedy+SAS and CQM averaged over ten examples with standard deviation.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof