## 4 Benchmarking
### Discussion  
We tested the basic VNS and B-VNS algorithms, using simulations on several QUBO and max-cut problems. QUBO problems from [PERSON] and [PERSON] as well as max-cut problems from [PERSON] and [PERSON] et al. were chosen for the test, since they are benchmarking standards. Thus, our simulations gave a good overview of the performance of the two algorithms. When solving the QUBO problems, the parameters of the two algorithms were set to be equivalent. Compared to the problem size \(n\), the parameters of both algorithms were set to very small values. We used \(k_{max}=0.02n\) and \(p_{max}=0.02\). As a result, the \(k_{max}\) in the VNS algorithm and the maximum \(\mu_{D}\) in the B-VNS algorithm only ranged from 1 to 10, while \(n\) ranged from 30 to 500. As the number of iterations was set at \(0.2n\), the number of iterations was in the range of 6 to 100. However, both the VNS and B-VNS algorithms were able to obtain the best-known values. The failure on just a few problems that was observed is understandable, due to the small values used for the parameters. The result would be improved, by enlarging the parameter values, i.e., by increasing \(k_{max}\) in the VNS algorithm, \(p_{max}\) in the B-VNS algorithm, and the number of iterations. However, increasing the parameter values may result in a longer computation time. For QUBO problems that apply equal conditions, the B-VNS algorithm was shown to be substantially faster than the VNS algorithm, for problems with sizes of less than 500. Moreover, the B-VNS algorithm was able to provide good solutions to all of the tested problems. Solving the standard QUBO problem, under equal conditions, showed that the B-VNS algorithm is able to match the VNS algorithm even better and faster. This trend, also, occurred in the tests for max-cut problems, even though the parameter settings were under non-equal conditions that risked the B-VNS algorithm being less thorough. However, the experiments show that reducing the thorough search behavior, by setting the chunk size \(C=0.9k_{max}\) in the B-VNS algorithm, did not lessen the solution quality, and it even ran substantially faster in all of the max-cut problems tested. Setting the parameter chunk size \(C\) to less than \(k_{max}\) has the risk of lowering the accuracy of the B-VNS algorithm. The characteristic of binomial distribution, which results in a pattern of randomly increasing distances, may, also, reduce accuracy. Small chunk size and binomial distribution can cause the algorithm to jump too far and miss some basins. However, our experiments and statistical analysis show that the B-VNS algorithm was, still, able to provide solutions, as satisfactorily as the VNS algorithm. Therefore, setting the chunk-size parameter to slightly less than \(k_{max}\) is advantageous for solution quality and efficiency. The flexible design, implemented by parameter chunk size \(C\), gives the B-VNS algorithm the potential to be faster than the VNS algorithm. Even though it is known that there is a risk of the B-VNS algorithm losing its thorough search characteristic, which may result in longer computation times, our experiments showed that this was not the case. The experiment results indicate that setting the B-VNS algorithm's parameters as equal to the VNS algorithm's parameters can avoid this risk.