## 4 Benchmarking
### Test on QUBO Problems  
The local search, as described in [37], was used in the tested algorithms. Figure 4 shows the pseudocode of the local search, used for QUBO problems. We applied equal conditions for the VNS and B-VNS algorithms. We aimed to reduce the values of all of the parameters, to reduce the computation time, while obtaining high-quality solutions. In the preliminary step, we used four problems for parameter tuning: two problems each, from Glover and Beasley, with sizes of 100. We started from larger parameter values  
Figure 4: Local search for QUBO [37]. and, gradually, reduced them. We found that \(k_{max}=0.02n\) was adequate for the VNS algorithm to obtain good solutions within short computation times. Hence, we set \(p_{max}=0.002\) on the B-VNS algorithm, to make it equal. The number of iterations was set at \(0.2n\), for both the VNS and B-VNS algorithms. The variable \(n\) was the problem size that was equal to the length of the solution vector. For the B-VNS algorithm, parameter chunk size \(C\) was set at \(0.02n\), so that it was equal to \(k_{max}\) in the VNS algorithm. All of these settings made an equal condition for the VNS and B-VNS algorithms. Table 1 shows the test results for Glover problems [11], while Table 2 showst test results for Beasley problems [37]. Tests regarding Glover problems show that the B-VNS algorithm was able to give the same good results as the VNS algorithmm in terms of objective function values. Both algorithms obtained the best-known value in almost all of the problems and only failed in the \(6b\) problem. The statistical analysis shows that the differences between the B-VNS and VNS algorithms were insignificant in all of the Glover test cases, as seen from the p-value greater than 0.05. The Mann-Whitney test (\(\alpha=0.05\)) was used because most samples did not satisfy the normality and homoscedasticity assumption. In terms of computation time, for tests on Glover problems, statistical analyses show that the B-VNS algorithm was faster than the VNS algorithm, specifically in problems with sizes up to 200. In problems with sizes of 500, the B-VNS algorithm was comparable to the VNS algorithm. The test results regarding the Beasley problems, also, show a similar trend to the tests on the Glover problems. However, the B-VNS algorithm could obtain all of the best-known values, while the VNS algorithm failed on problems \(bqp50\_1\), \(bqp100\_1\), and \(bqp250\_8\) (The notation \(bqpn\_m\) refers to the Beasley problem, which has size \(n\) and number \(m\)). Like the test on the Glover problem, as the differences were insignificant, both B-VNS and VNS algorithms were shown to be good algorithms, for use in solving Beasley problems. Statistical analyses for computation time, regarding Beasley problems with \(n<500\), show that the B-VNS algorithm was significantly faster than the VNS algorithm. The computation times of the two algorithms were only statistically the same in problems \(bqp50\_1\), \(bqp50\_5\), \(bqp50\_6\), \(bqp100\_8\), and \(bqp250\_7\). For problems where \(n>500\), the computation speeds for the B-VNS and VNS algorithms were comparable. All of the tests regarding QUBO problems under equal conditions show that the the B-VNS and VNS algorithms are good, as they reached most of the best-known values, with some exceptions for the VNS algorithm, due to its failures. Moreover, the B-VNS algorithm ran substantially faster than the VNS algorithm, particularly on problems with sizes less than 500. Therefore, the B-VNS algorithm is best suited for problems with sizes less than 500 and is comparable to the VNS algorithm for larger problems.