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Quantum Subroutines in Branch-Price-and-Cut for Vehicle Routing

Friedrich Wagner, Frauke Liers

TL;DR

The paper tackles the capacitated vehicle routing problem, a prototypical $NP$-hard problem, by embedding quantum subroutines into a classical branch-price-and-cut framework. It develops small, sparse $QUBO$ models for the pricing (CPCTSP) and strengthened separation (RCC) subproblems and uses quantum annealing as pricing and separation heuristics, while still allowing classical exact pricing and cuts to ensure optimality. Experimental results on real CVRP instances show that quantum annealing can reduce the number of expensive exact pricing calls and occasionally improve separations, but current hardware and overheads keep purely classical methods faster overall. The work demonstrates a viable pathway for leveraging quantum subroutines within established exact optimization algorithms and highlights the hardware requirements and future potential for practical quantum advantage in combinatorial optimization.

Abstract

Motivated by recent progress in quantum hardware and algorithms researchers have developed quantum heuristics for optimization problems, aiming for advantages over classical methods. To date, quantum hardware is still error-prone and limited in size such that quantum heuristics cannot be scaled to relevant problem sizes and are often outperformed by their classical counterparts. Moreover, if provably optimal solutions are desired, one has to resort to classical exact methods. As however quantum technologies may improve considerably in future, we demonstrate in this work how quantum heuristics with limited resources can be integrated in large-scale exact optimization algorithms for NP-hard problems. To this end, we consider vehicle routing as prototypical NP-hard problem. We model the pricing and separation subproblems arising in a branch-price-and-cut algorithm as quadratic unconstrained binary optimization problems. This allows to use established quantum heuristics like quantum annealing or the quantum approximate optimization algorithm for their solution. A key feature of our algorithm is that it profits not only from the best solution returned by the quantum heuristic but from all solutions below a certain cost threshold, thereby exploiting the inherent randomness is quantum algorithms. Moreover, we reduce the requirements on quantum hardware since the subproblems, which are solved via quantum heuristics, are smaller than the original problem. We provide an experimental study comparing quantum annealing to simulated annealing and to established classical algorithms in our framework. While our hybrid quantum-classical approach is still outperformed by purely classical methods, our results reveal that both pricing and separation may be well suited for quantum heuristics if quantum hardware improves.

Quantum Subroutines in Branch-Price-and-Cut for Vehicle Routing

TL;DR

The paper tackles the capacitated vehicle routing problem, a prototypical -hard problem, by embedding quantum subroutines into a classical branch-price-and-cut framework. It develops small, sparse models for the pricing (CPCTSP) and strengthened separation (RCC) subproblems and uses quantum annealing as pricing and separation heuristics, while still allowing classical exact pricing and cuts to ensure optimality. Experimental results on real CVRP instances show that quantum annealing can reduce the number of expensive exact pricing calls and occasionally improve separations, but current hardware and overheads keep purely classical methods faster overall. The work demonstrates a viable pathway for leveraging quantum subroutines within established exact optimization algorithms and highlights the hardware requirements and future potential for practical quantum advantage in combinatorial optimization.

Abstract

Motivated by recent progress in quantum hardware and algorithms researchers have developed quantum heuristics for optimization problems, aiming for advantages over classical methods. To date, quantum hardware is still error-prone and limited in size such that quantum heuristics cannot be scaled to relevant problem sizes and are often outperformed by their classical counterparts. Moreover, if provably optimal solutions are desired, one has to resort to classical exact methods. As however quantum technologies may improve considerably in future, we demonstrate in this work how quantum heuristics with limited resources can be integrated in large-scale exact optimization algorithms for NP-hard problems. To this end, we consider vehicle routing as prototypical NP-hard problem. We model the pricing and separation subproblems arising in a branch-price-and-cut algorithm as quadratic unconstrained binary optimization problems. This allows to use established quantum heuristics like quantum annealing or the quantum approximate optimization algorithm for their solution. A key feature of our algorithm is that it profits not only from the best solution returned by the quantum heuristic but from all solutions below a certain cost threshold, thereby exploiting the inherent randomness is quantum algorithms. Moreover, we reduce the requirements on quantum hardware since the subproblems, which are solved via quantum heuristics, are smaller than the original problem. We provide an experimental study comparing quantum annealing to simulated annealing and to established classical algorithms in our framework. While our hybrid quantum-classical approach is still outperformed by purely classical methods, our results reveal that both pricing and separation may be well suited for quantum heuristics if quantum hardware improves.

Paper Structure

This paper contains 21 sections, 2 theorems, 33 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our contribution.
  4. Structure.
  5. Preliminaries
  6. Quadratic unconstrained binary optimization.
  7. Quantum annealing.
  8. Branch-Price-and-Cut.
  9. The vehicle routing problem.
  10. Branch-Price-and-Cut Algorithm
  11. Set cover model.
  12. The pricing problem.
  13. Rounded capacity cuts.
  14. Solving the integer program.
  15. QUBO Models for Pricing and Separation
  16. ...and 6 more sections

Key Result

Lemma 4.1

Let $(x,y,w)$ be a binary vector such that $C^{\mathrm{CPCTSP}}_2(x,y,w)=0$ and let $(x',y',w')$ be a binary vector such that $C^{\mathrm{CPCTSP}}_2(x',y',w') > 0$. If then $C^{\mathrm{CPCTSP}}(x',y',w')>C^{\mathrm{CPCTSP}}(x,y,w)$.

Figures (3)

  • Figure 1: Schematic workflow of branch-price-and-cut. The algorithm builds a branch-and-bound tree (left). Each node represents a subset of solutions, where the root node corresponds to the entire solution set. The two children of a node divide its solution subset further into two disjoint subsets. For each node, we calculate a lower bound via column generation and cutting plane separation (right). As long as global upper and lower bound differ, the algorithm continues to branch (edges in the tree). If the local lower bound in a node exceeds the global upper bound, the node can be excluded from the search (nodes without children). We propose to use quantum annealing as a heuristic for pricing and separation.
  • Figure 2: (a): Schematic algorithm workflow for benchmarking pricing heuristics. We compare exclusively exact pricing via integer programming (IP) to simulated annealing (SA) and quantum annealing (QA). (b): Number of expensive exact pricings in dependence of the instance size $n$ (lower is better). (c): Average runtime per pricing problem in dependence of the instance size $n$. For QA, we also report the average time taken by the quantum processing unit (QPU).
  • Figure 3: (a): Schematic algorithm workflow for benchmarking separation heuristics. (b): Decrease in integrality gap for different instance sizes. Lower is better. (c): Total time spent on separation. For QA, we also report the average time taken by the quantum processing unit (QPU).

Theorems & Definitions (5)

  • Definition 2.1: CVRP
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof