Performance of the Extended Ising Machine for the Quadratic Knapsack Problem

The Quadratic Knapsack Problem is a well-known generalization of the classical 0-1 knapsack problem, in which any pair of items produces a pairwise profit if both are selected. Another relevant generalization of the knapsack problem is the Knapsack Problem with Setup, in which the items are partitioned into classes, the items of a class can only be inserted into the knapsack if the corresponding class is activated, and activating a class involves a setup cost and a setup capacity reduction.Despite a rich literature on these two problems, their obvious generalization, i.e., the Quadratic Knapsack Problem with Setup, was never investigated so far. We discuss applications, mathematical models, deterministic matheuristic algorithms, and computationally evaluate their performance.

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.

A merge process is proposed to engineer a multi-spin flip in an Ising machine. The merge process deforms the Hamiltonian (energy function) of the Ising model. We prove a theorem for the merge process and show that a single-spin flip in the deformed Hamiltonian is equivalent to a multi-spin flip in the original Hamiltonian. A merge process induces a transition within the subspace of feasible solutions. We propose a hybrid simulated annealing (SA) algorithm with the merge process. The hybrid algorithm outperforms the conventional SA algorithm, genetic algorithm, and tabu search in the binary quadratic knapsack problems (QKP) and the quadratic assignment problems (QAP). Finally, the hybrid merge process is used in a real Ising machine. The performance is improved in QKP and QAP instances. The merge process is generally applicable to existing Ising machine hardware because the deformed Hamiltonian keeps the format of the Ising model.

Simple solution methods for separable mixed linear and quadratic knapsack problem

The knapsack problem arises in real world applications, like transportation, manufacturing systems, finance, and supply chain management. In this paper, we investigate the use of a cooperative particle swarm optimization for solving the quadratic multiple knapsack problem. The standard swarm optimization is reenforced by using a local search procedure, where the swapping operator is introduced that combines items belonging to different bins (knapsacks) according to their critical items. The performance of the method is evaluated on benchmark instances of the literature, where its results are compared to the best available bounds available in the literature.

Parallel improved quantum inspired evolutionary algorithm to solve large size Quadratic Knapsack Problems

In this paper we have proposed a new hybrid approach combining artificial bee colony algorithm with a greedy heuristic and a local search for the quadratic knapsack problem. Quadratic knapsack problem belongs to traditional knapsack problem family and it is an extension of the well-known 0/1 knapsack problem. In this problem profits are also associated with pairs of objects along with individual objects. As this problem is an extension of the 0/1 knapsack problem, it is also $\mathcal{NP}$-Hard. Artificial bee colony algorithm is a new swarm intelligence technique inspired by foraging behavior of natural honey bee swarms. Performance of our algorithm on standard quadratic knapsack problem instances is compared with the other best heuristic techniques. Results obtained on these instances show that our hybrid artificial bee colony algorithm is superior to these techniques in many aspects.

Reoptimization in Lagrangian methods for the 0–1 quadratic knapsack problem

<span lang="EN">Quadratic Knapsack Problem is a variation of the knapsack problem that aims to maximize an objective function. The objective function in this case is quadratic. While the constraints used are binary and linear capacity constraints. The Whale Optimization Algorithm is a metaheuristic algorithm that can solve this problem. Therefore, this paper aims to find out the best solution to solve the Knapsack 0-1 Quadratic Problem using the Whale Optimization Algorithm so that its effectiveness and efficiency are known. Based on the research has been done, the algorithm is said to be effective because, from each experiment, the algorithm is always converging or towards maximum profit. Also, with the right parameters, the algorithm can achieve optimal results. It is said to be efficient because getting optimal profit does not require more time and iteration. The combination of item parameters and maximum iteration dramatically Affect the total value of profit and its running time. However, the addition of item parameter combinations is faster to achieve optimal than the maximum iteration parameter.</span>

Knapsack problem and its generalizations have been intensively studied during the last three decades with a rich literature of research reports. Over the years, surveys and reviews have been conducted mostly on the standard knapsack problems, namely, the single-constraint linear model. This paper reports the solution approaches developed for some non-standard knapsack problems with wide range of applications through a bibliographical review. The non-standard knapsack problems reviewed in this paper include the multidimensional knapsack problem, the multiple-choice knapsack problem, the 0–1 multiple knapsack problem, the quadratic knapsack problem, the maximin knapsack problem and the collapsing knapsack problem. Features of the solution approaches for each type of these non-standard knapsack problems and the computational experience as reported in the literature are summarized into several concise tables to provide a quick and broad reference for future researches.

We have developed a method to extend the functionality of the Ising machine, which currently has an energy function limited to a binary-quadratic form. The proposed method utilizes auxiliary variables dependent on the decision variables of the problem to be solved so that it can be accelerated by hardware parallel processing. Auxiliary variables add third-order or higher terms or rectified linear unit-type nonlinear functions to the binary-quadratic energy function of the Ising machine, modifying the value of the energy difference resulting from the reversal of the decision variable. To enable parallel computation, the computation of the energy change uses information that can be accessed locally by the decision neurons. We confirmed that the proposed method works for the 0/1 linear knapsack problem, the quadratic knapsack problem, the 3-SAT problem, and the random 3-XORST problem.

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems. The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of items

Knapsack problems — An overview of recent advances. Part I: Single knapsack problems

Solving the 0–1 Quadratic Knapsack Problem with a competitive Quantum Inspired Evolutionary Algorithm

The differences in performance among binary-integer encodings in an Ising machine, which can solve combinatorial optimization problems, are investigated. Many combinatorial optimization problems can be mapped to find the lowest-energy (ground) state of an Ising model or its equivalent model, the Quadratic Unconstrained Binary Optimization (QUBO). Since the Ising model and QUBO consist of binary variables, they often express integers as binary when using Ising machines. A typical example is the combinatorial optimization problem under inequality constraints. Here, the quadratic knapsack problem is adopted as a prototypical problem with an inequality constraint. It is solved using typical binary-integer encodings: one-hot encoding, binary encoding, and unary encoding. Unary encoding shows the best performance for large-sized problems.