Performance of the Extended Ising Machine for the Quadratic Knapsack Problem

2025-08-09

AI Summary

This paper addresses the challenge of efficiently solving the Quadratic Knapsack Problem (QKP) on conventional Ising machines. To overcome the limitations of the standard Quadratic Unconstrained Binary Optimization (QUBO) paradigm—which restricts variables to binary values and lacks native support for constraints—we propose the Extended Ising Machine (EIM) framework. EIM introduces real-valued dependent variables and explicit constraint modeling, significantly enhancing expressive power and modeling flexibility for constrained combinatorial optimization. By designing a QKP-specific constraint encoding scheme, EIM embeds the original problem directly into a continuous-discrete hybrid energy function, eliminating the need for large numbers of auxiliary variables. Experimental results on multiple benchmark instances demonstrate that EIM achieves superior solution quality and runtime efficiency compared to both standard Ising models and state-of-the-art baselines, including commercial exact solvers (e.g., Gurobi) and mainstream heuristic algorithms.

Technology Category

Constraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchMachine Learning: Quantum Machine Learning

Application Category

Economics, Online Markets and Human Computation: LLM based quality controls for crowd workGraph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The extended Ising machine (EIM) enhances conventional Ising models, which handle only binary quadratic forms by allowing constraints through real-valued dependent variables. We address the quadratic knapsack problem (QKP), hard to solve using Ising machines when formulated as a quadratic unconstrained binary optimization (QUBO). We demonstrated the EIM's superiority by comparing it with the conventional Ising model-based approach, a commercial exact solver, and a state-of-the-art heuristic solver for QKP.

Problem

• Solving quadratic knapsack problem with extended Ising machine
• Handling constraints through real-valued dependent variables
• Comparing performance against conventional and commercial solvers

Innovation

• Extended Ising Machine with real-valued variables
• Handles constraints beyond binary quadratic forms
• Solves quadratic knapsack problem effectively