We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability, and up to an additive constant with polynomial probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. These results imply polynomial time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.

翻译：我们证明了对于任意常数$k$，在$m \times n$网格图的$k$分量森林集合中，存在一个多项式比例的森林各分量具有相等的顶点数。这解决了Charikar、Liu、Liu和Vuong提出的猜想，并首次建立了基于生成树分布（该分布将每个$k$分区的权重定义为其$k$个片段各自生成树数量的乘积）进行（精确或近似）平衡网格图分区采样的多项式时间算法。该结果源于对网格图均匀随机生成树可切割成平衡片段的概率的精细分析。除网格图外，我们证明对于一大类类晶格图，能以常数概率实现任意乘法$(1 \pm \varepsilon)$常数范围内的平衡，并以多项式概率实现加法常数范围内的平衡。更一般地，我们证明由均匀生成树导出的分量能以常数概率近似给定Jordan曲线界定的平面区域任意划分。这些结果为类晶格图的近似平衡树加权分区采样提供了多项式时间算法。我们的结果对理解政治选区划分具有应用价值——在该场景中，不可分割的地理单元构成潜在图，需将其划分为$k$个人口平衡的连通子图。在此设定下，树加权分区具有有趣的几何性质，这极大地推动了相关采样方法的研究工作。