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Toward Practical Benchmarks of Ising Machines: A Case Study on the Quadratic Knapsack Problem

Kentaro Ohno, Tatsuhiko Shirai, Nozomu Togawa

TL;DR

The paper addresses practical benchmarks for Ising machines solving constrained combinatorial problems by focusing on the Quadratic Knapsack Problem (QKP). It introduces a simple two-stage post-processing scheme (repair followed by improvement) to convert infeasible Ising outputs into feasible solutions and then locally improve them, thereby relaxing the dependence on penalty encoding. Through extensive simulations and hardware experiments with the Amplify Annealing Engine, the method substantially improves solution quality and feasibility, achieving best-known results on many medium- and large-scale QKP instances and demonstrating competitive performance against specialized heuristics and solvers. The findings highlight a practical pathway to leverage Ising machines for constrained optimization and encourage broader benchmarking across problem classes and encoding strategies.

Abstract

Combinatorial optimization has wide applications from industry to natural science. Ising machines bring an emerging computing paradigm for efficiently solving a combinatorial optimization problem by searching a ground state of a given Ising model. Current cutting-edge Ising machines achieve fast sampling of near-optimal solutions of the max-cut problem. However, for problems with additional constraint conditions, their advantages have been hardly shown due to difficulties in handling the constraints. In this work, we focus on benchmarks of Ising machines on the quadratic knapsack problem (QKP). To bring out their practical performance, we propose fast two-stage post-processing for Ising machines, which makes handling the constraint easier. Simulation based on simulated annealing shows that the proposed method substantially improves the solving performance of Ising machines and the improvement is robust to a choice of encoding of the constraint condition. Through evaluation using an Ising machine called Amplify Annealing Engine, the proposed method is shown to dramatically improve its solving performance on the QKP. These results are a crucial step toward showing advantages of Ising machines on practical problems involving various constraint conditions.

Toward Practical Benchmarks of Ising Machines: A Case Study on the Quadratic Knapsack Problem

TL;DR

The paper addresses practical benchmarks for Ising machines solving constrained combinatorial problems by focusing on the Quadratic Knapsack Problem (QKP). It introduces a simple two-stage post-processing scheme (repair followed by improvement) to convert infeasible Ising outputs into feasible solutions and then locally improve them, thereby relaxing the dependence on penalty encoding. Through extensive simulations and hardware experiments with the Amplify Annealing Engine, the method substantially improves solution quality and feasibility, achieving best-known results on many medium- and large-scale QKP instances and demonstrating competitive performance against specialized heuristics and solvers. The findings highlight a practical pathway to leverage Ising machines for constrained optimization and encourage broader benchmarking across problem classes and encoding strategies.

Abstract

Combinatorial optimization has wide applications from industry to natural science. Ising machines bring an emerging computing paradigm for efficiently solving a combinatorial optimization problem by searching a ground state of a given Ising model. Current cutting-edge Ising machines achieve fast sampling of near-optimal solutions of the max-cut problem. However, for problems with additional constraint conditions, their advantages have been hardly shown due to difficulties in handling the constraints. In this work, we focus on benchmarks of Ising machines on the quadratic knapsack problem (QKP). To bring out their practical performance, we propose fast two-stage post-processing for Ising machines, which makes handling the constraint easier. Simulation based on simulated annealing shows that the proposed method substantially improves the solving performance of Ising machines and the improvement is robust to a choice of encoding of the constraint condition. Through evaluation using an Ising machine called Amplify Annealing Engine, the proposed method is shown to dramatically improve its solving performance on the QKP. These results are a crucial step toward showing advantages of Ising machines on practical problems involving various constraint conditions.
Paper Structure (27 sections, 2 theorems, 17 equations, 8 figures, 21 tables, 1 algorithm)

This paper contains 27 sections, 2 theorems, 17 equations, 8 figures, 21 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ising machines
  4. Quadratic Knapsack Problem
  5. Challenges in Ising Machines Solving QKP
  6. Proposed Method
  7. Post-processing Algorithm on QKP
  8. Simulation Experiments
  9. Computational Set-up
  10. Results
  11. Observations from Tuning of Penalty Coefficients
  12. Results on Best Optimality Gap
  13. Results on Processing Time
  14. Dependency on Encoding Methods
  15. Analysis of Optimal Penalty Coefficients
  16. ...and 12 more sections

Key Result

Proposition 1

For a QKP instance defined as (eq:qkp), an optimum is attained by a solution $x \in \{0,1\}^n$ satisfying $C-\max_i w_i < \sum_{i=1}^n w_i x_i \le C$.

Figures (8)

  • Figure 1: Conceptual figure of effect of two-stage post-processing. "Raw solutions" denote outputs of Ising machines, which are often infeasible when penalty coefficient $\lambda$ is small (dashed line on "Critical domain"). Tuning of $\lambda$ typically involves finding $\lambda_\mathrm{critical}$ which achieves best trade-off between feasibility and objective. Repair procedure for infeasible solutions enables us to obtain feasible solutions even for smaller $\lambda$. Improvement procedure further enhances feasible solutions with local operations. Optimal penalty coefficient $\lambda_\mathrm{optimal}$ is found to be much robust to choice of encoding methods for inequality constraint, in contrast to $\lambda_\mathrm{critical}$ which heavily depends on encoding methods (see Section \ref{['sec:simulation']}).
  • Figure 2: Optimality gap (line graph) and number of instances on which feasible solutions are obtained with SA (bar chart) for each problem size $n$ of QKP instances. Optimality gap for SA and SA-I is plotted only for $\lambda$ producing feasible solutions on all instances. By combining repair and improvement procedures, SA-RI achieves smaller optimality gap than greedy method.
  • Figure 3: Performance comparison among various encoding methods of inequality constraint on 100 medium-sized QKP instances. (a)(b) Choice of encoding methods controls trade-off between rates of feasible solutions and objective values. (c) Solving performance of proposed method is much less dependent on choice of encoding methods.
  • Figure 4: Processing time of processes in AE-RI. Lines are fitted with log-log regression. The execution time of AE is set to $O(n)$ and the other processes empirically take $O(n^2)$ runtime.
  • Figure 5: Optimality gap for each problem size $n$ of QKP instances. Optimality gap for SA is plotted only for $\lambda$ producing feasible solutions on all instances. SA-RF denotes SA-R followed by fill-up operation, which produces locally optimal solutions. Fill-up operation improves solutions of SA-R particularly around $\lambda=2$, which is optimal penalty coefficient for SA-RI.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 1: cf. billionnet1996linear
  • Proposition 2
  • proof