# 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
Suppose there exists a superior solution \(O=\{o_{1},\,o_{2},\,\ldots,\,o_{m}\}\) with strictly greater weight, i.e., \(w(O)>w(G)\). 3. By leveraging the exchange property of matroids, element substitution operations can be performed to demonstrate that the weight of \(G\) is at least that of \(O\), thereby contradicting the initial assumption. \(\square\)

### Correlation Derivation Between Reverse Path and Forward Selection

1. Proposition: Under the matroid structure, the reverse path (deriving local selections by backward induction from the maximal independent set) shares an identical global optimal solution with the forward greedy selection. Proof. : 1. Forward Greedy Selection: Construct solution \(G=\{g_{1},g_{2},\,\ldots,g_{k}\}\) by sequentially selecting elements in descending order of weights. 2. Reverse Path Construction: Initialize with universal set S. Remove elements in ascending order of weights while preserving the current maximum-weight independent set. The retained set ultimately becomes \(G^{\prime}=\{g^{\prime}1,g^{\prime}2,\,\ldots,g^{\prime}k\}\). 3. Equivalence Proof: By heredity property of matroids, removing low-weight elements preserves independence. Each removal operation in a reverse path corresponds to omitting low-weight elements in forward selection. Therefore, \(G^{\prime}=G\), demonstrating identical global optimal solutions from both approaches. 4. Formal Derivation: Let the weight function be \(w\), universal set \(S\), and maximum-weight independent set \(G\). For each step in the reverse path, the removed element \(e\) satisfies: \[e=\operatorname*{argmin}_{e\in S\backslash G^{\prime}}w(e)\] (A4)By matroid exchange property, the remaining set preserves maximal weight independence after removing \(e\). \(\square\)

### Conclusions

In combinatorial optimization problems involving matroids or those possessing optimal substructures, the reverse path of the greedy algorithm exhibits a rigorous global optimality correlation with its forward selection process. This is fundamentally underpinned by the exchange and hereditary properties of matroids, as well as the optimal substructure property inherent in dynamic programming. Such conclusions are rigorously supported in classical algorithmic literature and have been validated through practical implementations, e.g., [PERSON]'s algorithm [PERSON] (1999). ## References

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