## 4 Conclusions  
In this study, we explored the quadratic unconstrained binary optimization (QUBO) problem, which produced significant findings. We established a necessary and sufficient condition for the local optimality of an \(r\)-flip search after a 1-flip search had already achieved local optimality. Our computational experiments demonstrated a substantial reduction in candidate options for \(r\)-flip implementation. The new \(r\)-flip strategy efficiently solved extremely large QUBO instances within 600 s, outperforming MST2 in terms of the time taken to reach the best-known solutions in benchmark instances. These results are particularly promising for implementing variable neighborhood strategies on extensive problems or sparse matrices.  
Similar to other quadratic unconstrained binary optimization (QUBO) methods, our approach is subject to scalability constraints. The size of the Q matrix, representing the number of variables in the problem, is inherently restricted by the memory limitations of the computer system. In practical terms, our methodology exhibits robust performance, successfully tackling problem instances of up to 200,000 by 200,000 in dense matrices, with the potential to extend its capabilities to 500,000 by 500,000 in sparse matrices. This underscores the versatility and efficiency of our approach within the specified computational constraints, demonstrating its applicability to a wide range of problem sizes and structures.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & \multicolumn{2}{c}{**MST2**} & \multicolumn{2}{c}{**1-Flip**} & \multicolumn{2}{c}{**2-Flip**} \\ \cline{2-7}
**ID** & **AT (s)** & **DT** & **AT (s)** & **DT** & **AT (s)** & **DT** \\ \hline p5000\_4 & 279.4 & 5.987 & 270.3 & 48.842 & 143.0 & 0 \\ p5000\_5 & 287.1 & 9.234 & 194.5 & 94.641 & 172.7 & 92.268 \\ p6000\_1 & 424.8 & 4.555 & 152.9 & 95.641 & 145.9 & 46.657 \\ p6000\_2 & 498.0 & 0 & 142.5 & 97.375 & 73.8 & 122.070 \\ p6000\_3 & 252.3 & 5.571 & 248.0 & 51.129 & 318.1 & 61.672 \\ p7000\_1 & 344.5 & 0.153 & 272.5 & 93.675 & 441.5 & 12.974 \\ p7000\_2 & 587.0 & 0 & 115.0 & 0 & 265.1 & 76.694 \\ p7000\_3 & 109.0 & 0 & 84.5 & 39.472 & 131.1 & 55.401 \\ p8000\_1 & 548.6 & 1.398 & 251.3 & 63.346 & 141.0 & 0 \\ p8000\_2 & 145.4 & 11.829 & 258.3 & 56.352 & 300.7 & 56.001 \\ p8000\_3 & 514.3 & 11.365 & 368.8 & 43.667 & 417.3 & 12.797 \\ p30000\_1 & 572.0 & 1.365 & 598.0 & 0 & 592.0 & 0 \\ p30000\_2 & 581.5 & 1.526 & 580.0 & 0 & 571.0 & 0 \\ p3000\_3 & 586.5 & 2.773 & 599.0 & 0 & 590.0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 7: _Cont._  
**Author Contributions:** Conceptualization, B.A. and H.W.; methodology, B.A.; validation, B.A., H.W. and L.S.S.; formal analysis, H.W.; investigation, H.W.; data curation, H.W.; writing--original draft preparation, B.A.; writing--review and editing, H.W. and L.S.S.; visualization, B.A.; All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** The instance data and solutions will be made available upon request.  
**Conflicts of Interest:** The authors declare no conflict of interest.