We consider the probability that a spanning tree chosen uniformly at random from a graph can be partitioned into a fixed number $k$ of trees of equal size by removing $k-1$ edges. In that case, the spanning tree is called {\em splittable}. Splittable spanning trees are useful in algorithms for sampling {\em balanced forests}, forests whose components are of equal size, and for sampling partitions of a graph into components of equal size, with applications in redistricting, network algorithms, and image decomposition. Cannon et al.~recently showed that spanning trees on grid and grid-like graphs on $n$ vertices are splittable into $k$ equal sized components with probability at least $n^{-2k}$, leading to the first rigorous sampling algorithm for balanced forests in any class of graphs. Focusing on the complementary case of dense random graphs, we show that random spanning trees have inverse polynomial probability of being splittable; specifically, a random spanning tree is splittable with probability at least $n^{(-k/2)}$ for both the $G(n,p)$ and $G(n,m)$ models when $p = Ω(1/\log n)$, giving the first dense class of graphs where partitions of equal size can be sampled efficiently. In addition, we present an infinite family of graphs with properties that have been conjectured to ensure splittability (i.e., Hamiltonian subgraphs of the triangular lattice) and prove that random spanning trees are not splittable with more than exponentially small probability. As a consequence, we show that a family of widely-used Markov chain algorithms for sampling equal-size partitions will fail on this family of graphs if their state spaces are restricted to equal-size partitions. Moreover, we show these algorithms will be inefficient if their state spaces are generalized to include any unbalanced partitions, suggesting barriers for sampling balanced partitions in sparse graphs.

翻译：本文研究从图中均匀随机选取的生成树，通过移除$k-1$条边后能否被划分为$k$棵规模相等的子树。满足此条件的生成树称为{\em 可分割生成树}。可分割生成树在采样{\em 平衡森林}（即各连通分量规模相等的森林）以及将图划分为等规模分区的算法中具有重要应用，其应用场景包括选区重划、网络算法和图像分解等领域。Cannon等人近期证明，在具有$n$个顶点的网格及类网格图中，生成树能以至少$n^{-2k}$的概率被分割为$k$个等规模分量，这为在任何图类中实现平衡森林的严格采样算法提供了首个理论依据。针对稠密随机图这一互补情形，本文证明随机生成树具有逆多项式量级的可分割概率：具体而言，在$p = Ω(1/\log n)$条件下，$G(n,p)$与$G(n,m)$模型中的随机生成树至少以$n^{(-k/2)}$的概率可分割，这首次为稠密图类中等规模划分的高效采样提供了理论支持。此外，我们提出了一类具有推测可分割性质（即三角晶格的哈密顿子图）的无限图族，并证明其随机生成树的可分割概率不超过指数小量。由此表明，若将状态空间限制在等规模划分，一类广泛使用的等规模划分采样马尔可夫链算法在该图族上必然失效。进一步地，我们证明即使将状态空间扩展至包含非平衡划分，这些算法仍将保持低效，这揭示了稀疏图中平衡划分采样面临的理论障碍。