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Lecture notes in computer science

Solving Quadratic Knapsack Problem with Biased Quantum State Optimization Algorithm

Huy Phuc Nguyen Ha, Viết Hùng Nguyễn, Anh Son Ta

Published Jan 1, 2024• 1 citation
Publisher
DOI
10.1007/978-3-031-62912-9_26

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The binary quadratic knapsack problem (QKP) aims at optimizing a quadratic cost function within a single knapsack. Its applications and difficulty make it appealing for various industrial fields. In this paper we present an efficient strategy to solve the problem by modeling it as an Ising spin model using an Ising machine to search for its ground state which translates to the optimal solution of the problem. Secondly, in order to facilitate the search, we propose a novel technique to visualize the landscape of the search and demonstrate how difficult it is to solve QKP on an Ising machine. Finally, we propose two software solution improvement algorithms to efficiently solve QKP on an Ising machine.
Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the “Quantum Tree Generator” (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.
Abstract The quadratic knapsack problem(QKP) is a typical combinatorial optimization problem. It is arisen in many optimization fields, but there exist no pseudo-polynomial time algorithm to solve it. In this paper, we propose an angle-expressed quantum evolutionary algorithm to solve QKP. In this algorithm, the qubits are expressed in the angle and initialized according to the value densities of their corresponding items, the rotation angle of Q-gate is determined through an analytical formula, and Hε gate is used to prevent from premature convergence. For infeasible solution, a dynamic value density is introduced to choose item to be selected or dropped. Finally, experiment demonstrates the effectiveness of the algorithm.