# Improved Branch-and-Bound Antenna Selection Algorithm for Massive MIMO
## 3 Efficient Antenna Selection via Enhanced Branch-and-Bound Optimization
### Branch-and-Bound Antenna Selection
The computational complexity of the I-BAB algorithm was evaluated by measuring the number of nodes accessed during the search process. We set \(N_{t}=d=4\) and gradually increased \(N_{r}\) from 8 to 256. As benchmarks, we also evaluate the exhaustive search algorithm and the greedy selection algorithm, representing the worst-case and best-case scenarios, respectively. The results are shown in Figure 5. Since the exhaustive search algorithm lacks pruning operations, the number of nodes it explores grows exponentially with an increasing \(N_{r}\). In contrast, the greedy selection algorithm computes only d nodes, resulting in significantly lower computational complexity. As illustrated in Figure 5, the horizontal coordinate represents the number of receiving antennas \(N_{r}\) and the vertical coordinate represents the number of nodes visited during the search; the number of nodes visited by the I-BAB algorithm is approximately one-tenth of that required by the BAB algorithm. This highlights the efficiency of the I-BAB-based antenna selection method in reducing the search complexity compared to the traditional BAB algorithm. However, its computational cost remains notably higher than that of the greedy search algorithm. Furthermore, when \(N_{r}\) increases from 8 to 256, the computational complexity of both the BAB and I-BAB algorithms grows at a moderate rate, increasing by nearly two orders of magnitude. In contrast, the exhaustive search algorithm exhibits a much steeper rise in complexity, reaching approximately five orders of magnitude. This confirms that in large-scale antenna configurations, the I-BAB algorithm offers a significant advantage in computational efficiency. Next, we analyze the algorithmic complexity from the perspective of the actual runtime. As previously mentioned, the number of selected antennas in the proposed I-BAB algorithm is no longer confined to a specific value but can be multiple values within a range. Through a single execution, the I-BAB algorithm can obtain the optimal subset corresponding to each value in this range, thereby eliminating the limitation of conventional BAB algorithms that require multiple executions for different values of \(d\). We set \(N_{r}=16\). For the BAB algorithm, we employed MATLAB R2023a to obtain the runtime at \(d=2\), 4, and 8, respectively. For the I-BAB algorithm, we configured \(d=\) [2, 4, 8] and similarly acquired corresponding runtime measurements using MATLAB. The execution

Figure 5: Complexity comparison, \(N_{t}=4\), \(d=4\) and \(\rho=15\) dB. time of both algorithms is systematically presented in Table 3. As can be observed from Table 3, the BAB algorithm requires three executions for three distinct d values. When \(d=2\), the execution time is measured at \(1.967\times 10^{-3}\) s, whereas for \(d=8\), it increases to approximately \(1.659\times 10^{-1}\) s. This clearly demonstrates that the execution time of the BAB algorithm exhibits a significant upward trend with an increasing \(d\); it is evident that the execution time of the BAB algorithm demonstrates a significant growth trend as the \(d\)-value increases. Meanwhile, we calculated that the total computation time required for three runs of the BAB algorithm is approximately \(1.73\times 10^{-1}\) s. In contrast, the I-BAB algorithm only requires a single execution to obtain optimal antenna subsets corresponding to three different \(d\) values, with a significantly reduced computation time of merely \(3.0619\times 10^{-2}\) s. Compared with the BAB algorithm, the marked reduction in runtime of I-BAB not only demonstrates its lower computational complexity but also highlights a crucial advantage: the enhanced operational flexibility in practical implementations. Finally, we investigate the performance of the I-BAB algorithm under varying SNR conditions. As shown in Figure 6, the horizontal coordinate represents the signal-to-noise ratio, and the vertical coordinate represents the channel capacity; the channel capacity exhibits a monotonic increase with the SNR.