Motivated by applications in political districting, we consider the task of partitioning the n vertices of a planar graph into k connected components. We propose an extended formulation for this task that has two desirable properties: (i) it uses just O(n) variables, constraints, and nonzeros, and (ii) it is perfect. To explore its ability to solve real-world problems, we apply it to a political districting problem in which contiguity and population balance are imposed as hard constraints and compactness is optimized. Computational experiments show that, despite the model’s small size and integrality for connected partitioning, the population balance constraints are more troublesome to effectively impose. Nevertheless, we share our findings in hopes that others may find better ways to impose them.

受政治选区应用的推动，我们考虑将平面图的n 个顶点划分为k 个连通分量的任务。我们为该任务提出了一个扩展公式，它具有两个理想的属性：(i) 它仅使用O ( n ) 变量、约束和非零值，(ii) 它是完美的。为了探索其解决现实世界问题的能力，我们将其应用于政治选区问题，其中邻接性和人口平衡被强加为硬约束，并优化紧凑性。计算实验表明，尽管模型规模较小且连通划分完整，但种群平衡约束的有效施加比较困难。尽管如此，我们还是分享我们的发现，希望其他人能找到更好的方法来实施它们。