The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges. This paper presents Integer Programming (IP) approaches for MASTP, introduces pre-processing techniques to reduce the size of the formulations, and characterizes valid inequalities for reinforcing its Linear Programming Relaxation bounds. Several Branch-and-cut (BC) algorithms exploiting such ideas are introduced. Additionally, we also apply Benders Decomposition to one of these formulations. An accompanying BC method, that separates Benders optimality cuts, is also introduced. Aiming to save linear programming re-optimization times, that algorithm makes use of an early branching strategy. Given the set of valid inequalities used in the polyhedral representation of the problem and the best available upper bound for the optimal cost, it detects if the node cannot be pruned by bounds and then stops the cutting plane generation, in order to branch. Our algorithms manage to solve instances with up to only 15 vertices, suggesting thus that MASTP is hard to solve in practice, at least with the currently available methods. Thanks to the early branching strategy, the Benders based BC method obtained the best computational results, by far. It solved more instances to proven optimality than the other algorithms (31 out of 45 cases) and it is about three times faster than the second best performing algorithm.

最小面积生成树问题（MASTP）是根据完整的无向图G定义的，其中每个顶点都代表二维欧几里得平面中的一个点。与每个边缘相关联的是，在其中点处放置了一个圆盘，其直径与边缘的长度匹配。MASTP寻找G的生成树这样可以最大程度地减少与磁盘边缘相关联的磁盘联合中的面积。本文介绍了用于MASTP的整数编程（IP）方法，介绍了预处理技术以减小公式的大小，并描述了有效的不等式以增强其线性规划弛豫范围。介绍了几种利用这种思想的分支剪切（BC）算法。此外，我们还将Benders分解应用于这些公式之一。还介绍了一种相伴的BC方法，该方法将Benders最优切割分开。为了节省线性编程的重新优化时间，该算法利用了早期的分支策略。给定问题的多面体表示中使用的一组有效不等式以及最佳成本的最佳可用上限，它检测是否无法通过边界修剪该节点，然后停止生成切割平面以进行分支。我们的算法设法解决最多只有15个顶点的实例，因此表明MASTP在实践中很难解决，至少使用当前可用的方法很难解决。得益于早期的分支策略，到目前为止，基于Benders的BC方法获得了最佳的计算结果。它比其他算法（45个案例中的31个）解决了更多实例，证明了最优性，并且比性能第二好的算法快约三倍。得益于早期的分支策略，到目前为止，基于Benders的BC方法获得了最佳的计算结果。它比其他算法（45个案例中的31个）解决了更多实例，证明了最优性，并且比性能第二好的算法快约三倍。得益于早期的分支策略，到目前为止，基于Benders的BC方法获得了最佳的计算结果。它比其他算法（45个案例中的31个）解决了更多实例，证明了最优性，并且比性能第二好的算法快约三倍。