The formation trajectory planning using complete graphs to model collaborative constraints becomes computationally intractable as the number of drones increases due to the curse of dimensionality. To tackle this issue, this paper presents a sparse graph construction method for formation planning to realize better efficiency-performance trade-off. Firstly, a sparsification mechanism for complete graphs is designed to ensure the global rigidity of sparsified graphs, which is a necessary condition for uniquely corresponding to a geometric shape. Secondly, a good sparse graph is constructed to preserve the main structural feature of complete graphs sufficiently. Since the graph-based formation constraint is described by Laplacian matrix, the sparse graph construction problem is equivalent to submatrix selection, which has combinatorial time complexity and needs a scoring metric. Via comparative simulations, the Max-Trace matrix-revealing metric shows the promising performance. The sparse graph is integrated into the formation planning. Simulation results with 72 drones in complex environments demonstrate that when preserving 30\% connection edges, our method has comparative formation error and recovery performance w.r.t. complete graphs. Meanwhile, the planning efficiency is improved by approximate an order of magnitude. Benchmark comparisons and ablation studies are conducted to fully validate the merits of our method.

翻译：利用完全图对协同约束进行建模的编队轨迹规划，随着无人机数量的增加会因维度灾难而变得计算棘手。为解决这一问题，本文提出了一种面向编队规划的稀疏图构建方法，以实现更好的效率-性能权衡。首先，设计了一种针对完全图的稀疏化机制，以确保稀疏化后的图具有全局刚性，这是唯一对应几何形状的必要条件。其次，构建了良好的稀疏图以充分保留完全图的主要结构特征。由于基于图的编队约束由拉普拉斯矩阵描述，稀疏图构建问题等价于子矩阵选择，其具有组合时间复杂度并需要评分指标。通过对比仿真，Max-Trace矩阵揭示指标展现出优越性能。我们将稀疏图集成到编队规划中。在复杂环境中对72架无人机的仿真结果表明，在保留30%连接边的情况下，我们的方法相较于完全图具有可比的编队误差和恢复性能，同时规划效率提升约一个数量级。通过基准对比和消融实验充分验证了本方法的优势。