## 1 Introduction  
Combinatorial optimization has attracted a good deal of attention, as it has many applications in various fields [1]. However, it is not always easy to solve combinatorial optimization problems, especially in some cases, classified as NP-hard problems. In this kind of situation, the use of the approximation method is a reasonable option [2]. One of the approximation methods that has attracted a great deal of attention in modern optimization is the metaheuristic method [3], which is designed to obtain good solutions within a reasonable time frame [4]. Even though it is not easy to prove that a solution obtained using the metaheuristic method is a global optimum [5], the results are, often, very close to the global optimum. Depending on the case, a combinatorial problem can be formulated in various ways, so that it can be be easily solved. One method is the use of a Boolean or binary vector, which, despite its simplicity, is a compelling method for solving many combinatorial problems. A specific binary formulation forms a major optimization problem category, called \"quadratic unconstrained binary optimization (QUBO)\", or \"Ising model optimization\" in some of the literature [6]. In the QUBO problem, given \(\mathbf{Q}=(q_{ij})\) is a symmetric \(n\)-square matrix of coefficients, the objective is to maximize the function:  
\[f(\mathbf{X})=\mathbf{X}^{\mathbf{T}}\mathbf{Q}\mathbf{X}=\sum_{i=1}^{n}\sum_{j=1}^{n}q_{ij}x_{i}x_ {j} \tag{1}\]where \(\mathbf{X}=(x_{i})\) is an \(n\)-dimensional vector of binary variables, i.e., \(x_{i}\in\{0,1\}\) for \(i=1,2,...,n\). QUBO problems are categorized as NP-hard problems [7], and their decision form is NP-complete [6]. The implementation of the QUBO formulation can be found in many combinatorial problems, e.g., graph coloring, partition, and maximum cut (max-cut) [6], which were included in [PERSON]'s original 21 NP-complete problems [8]. More applications of QUBO were described by [PERSON] and [PERSON] [9]. It is as difficult to solve QUBO problems as it is to solve other NP-hard problems. Many metaheuristic algorithms have been devised to solve them, based on the requirement that they are solved within a reasonable amount of time. Examples of these are simulated annealing [10], tabu search [11], genetic algorithm [12], and local search [13] algorithms. One particular metaheuristic algorithm that uses local search is the variable neighborhood search (VNS) algorithm [14]. The VNS algorithm can solve various problems, including QUBO problems. As a single-trajectory-based algorithm, the VNS algorithm has the advantage that it is resource-efficient. Furthermore, its memory requirements are relatively low, compared to population-based metaheuristic algorithms. Therefore, the VNS algorithm is suitable for use in solving large-scale problems and does not need large memory allocations. It is a simple concept, and the VNS algorithm is, often, used in its original form, a modified form, or hybridized with another algorithm [15; 16]. The VNS algorithm has been shown to perform well in various problems. Its applications include the max-cut combinatorial problem [17; 18], the scheduling problem [19; 20], the layouting problem [21], the vehicle routing problem [22], and the multiprocessor scheduling problem with communication delays [23]. It has, also, been used to tune proportional-integral-derivative controllers in cyber-physical systems [24]. Not only has the VNS algorithm been applied in numerous ways, but it has, also, been, often, used in combination with other algorithms to obtain combined performance. It is possible to hybridize the VNS algorithm with other metaheuristic algorithms, such as with the genetic algorithm [25], the particle swarm optimization algorithm [26], the migrating birds optimization algorithm [27], and the simulated annealing algorithm [28]. The VNS algorithm can also be implemented in parallel programming, by using a graphical processing unit [29] to leverage its performance.