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. This resolves a conjecture of Charikar, Liu, Liu, and Vuong. It also 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 has 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：我们证明在$m \times n$网格图中，所有$k$分支森林集合中多项式比例的子集具有每个分支顶点数相等的性质。这解决了Charikar、Liu、Liu与Vuong提出的猜想，同时首次确立了（精确或近似）按生成树分布采样平衡网格图划分的可证明多项式时间算法——该分布通过各分块生成树数量的乘积对每个$k$划分进行加权。该结果对理解政治选区划分具有应用价值：其底层由不可分割地理单元构成图，需划分为$k个$人口均衡的连通子图。在此场景下，树加权划分展现出有趣的几何性质，这推动了大量针对其采样方法的研究工作。