## 3 Proposed Neighborhood Structure
The distribution of the obtained neighboring point \(\mathbf{X}^{\prime}\) corresponds to the probability density function of the binomial distribution, as shown in Figure 2. The concept is advantageous because the distance of neighboring points \(\mathbf{X}^{\prime}\) can be estimated, even though they are random. The proposed neighborhood structure is illustrated in Figure 2(a). Note that the illustration is a simple version, as \(\mathbf{X}^{\prime}\) can be obtained in any direction. The neighborhood expansion corresponds to flip probability \(p_{c}\), following the variable chunk \(c\). The distance between neighboring point \(\mathbf{X}^{\prime}\) and \(\mathbf{X}\) will be random, even though it will follow the characteristic of binomial distribution. In contrast, in the basic VNS algorithm, the neighborhood structure can be illustrated as an expanded disc, as shown in Figure 2(b). In the basic VNS algorithm, the expansion corresponds to variable \(k\) and results in \(d(\mathbf{X}^{\prime},\mathbf{X})\) being exactly \(k\). The complete implementation of our proposed method for solving QUBO problems is similar to that of the VNS algorithm, except for differences in the neighborhood construction. Unlike the basic VNS algorithm, we use parameter \(p_{max}\) to control the distance. However, just like VNS, neighborhood structure change should be limited. Therefore, we use parameter chunk size C. Thus, we only change the construction of the neighborhood structure. For simplicity, we call our proposed algorithm \"B-VNS\", while \"VNS\" refers to the basic VNS algorithm [14; 17; 31; 34]. Changing the construction of the neighborhood structure will change the behavior. The B-VNS algorithm is more flexible than the VNS algorithm. In terms of \(k_{max}\) in the VNS algorithm, the B-VNS algorithm can reach almost the same neighborhood structure, by setting up the parameter \(p_{max}\) that  
\[\mu_{D}=np_{max}\equiv k_{max}. \tag{9}\]  
However, the B-VNS algorithm is more flexible than the VNS algorithm. The chunk size \(C\) can be set as equal to \(k_{max}\), to give an almost equal condition. However, the chunk size \(C\) can be set as either larger or less than \(k_{max}\). Setting the chunk size \(C\) to less than \(k_{max}\) may have the potential to speed up the computation time. In addition to the potential advantages of B-VNS, there are, also, potential weaknesses. Like the Jump VNS algorithm, in the B-VNS algorithm, thorough search behavior may be lost. The VNS algorithm, as described by [PERSON] and [PERSON] [5; 31] and [PERSON] et al. [17], performs a thorough search, by starting with the nearby neighborhood structure and slowly expanding it,when the local search fails to improve the solution. In contrast, in the proposed algorithm, a random pattern of the distance of the neighborhood structures is seen. This random characteristic can be advantageous, as it can increase the exploration search. However, a drawback is that it causes B-VNS to miss basins that the global optimum may be in. Consequently, this random characteristic has a possibility of incurring a longer searching time than the basic VNS algorithm.