The Quadratic Knapsack Problem (QKP) deals with maximizing a quadratic objective function subject to given constraints on the capacity of the Knapsack. We assume all coefficients to be non-negative and all variables to be binary. Solution to QKP generalizes the problem of finding whether a graph contains a clique of given size. We propose in this paper a Novel Quantum Evolutionary Algorithm (NQEA) for QKPs. These algorithms are general enough and can be used for similar subsection of problems. We report in this paper solutions which lie in less than 1% of the optimal solutions. We also show that our algorithm is scalable to much larger problem sizes and is capable of exploiting the search space to its maximum.
The Quadratic Knapsack Problem (QKP) deals with maximizing a quadratic objective function subject to given constraints on the capacity of the Knapsack. We assume all coefficients to be non-negative and all variables to be binary. Solution to QKP generalizes the problem of finding whether a graph contains a clique of given size. We propose in this paper a Novel Quantum Evolutionary Algorithm (NQEA) for QKPs. These algorithms are general enough and can be used for similar subsection of problems. We report in this paper solutions which lie in less than 1% of the optimal solutions. We also show that our algorithm is scalable to much larger problem sizes and is capable of exploiting the search space to its maximum.
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.
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.
In this paper, we propose a novel quantum-inspired evolutionary algorithm, called NQEA, for solving combinatorial optimization problems. NQEA uses a new Q-bit update operator to increase the balance between the exploration and exploitation of the search space. In the operator, first, the Q-bits of each individual in the population are updated based on the personal best measurement of that individual and the best measurement of current generation. Then, a restriction is applied to each Q-bit to prevent the premature convergence of its values. The results of experiments on the 0-1 knapsack and NK-landscapes problems show that NQEA performs better than a classical genetic algorithm, CGA, and two quantum-inspired evolutionary algorithms, QEA and vQEA, in terms of convergence speed and accuracy.
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.
In this paper, a generalization of the original Quantum-Inspired Evolutionary Algorithm (QIEA): the Generalized Quantum-Inspired Evolutionary Algorithm (GQIEA) is proposed. Like QIEA, GQIEA is also based on the quantum computing principle of superposition of states, but extending it not only to be used for binary values {0, 1}, but for any finite set of values {1, n}. GQIEA, as any other quantum inspired evolutionary algorithm, defines an own quantum individual, an evaluation function and population operators. As in QIEA, GQIEA also defines a generalized Q-gate operator, which is a variation operator to drive the individuals toward better solutions. To demonstrate its effectiveness and applicability, the proposal will be applied to the Knapsack Problem (KP), a classic combinatorial optimization problem. Results show that GQIEA has a good performance even with a small population.
Knapsack problem is a typical combinatorial optimization problem, which has attracted wide attention because of its simple description, difficult solution and wide application background. For the knapsack problem, we proposed an improved quantum evolutionary algorithm. In this algorithm, the qubits are initialized according to the value densities of their corresponding items, the angle of Q-gate is determinated through direct comparison between the quantum chromosome and the optimal solution, random crossover method is taken to share information among chromosomes, and Hϵ gate is used to prevent from premature convergence. Finally, experiment demonstrates the effectiveness of the algorithm.