Splittable Spanning Trees and Balanced Forests in Dense Random Graphs

TL;DR

The paper analyzes the likelihood that a uniformly random spanning tree is splittable into $k$ equal-sized components, enabling efficient sampling of balanced forests. It develops a formal framework based on the spanning-tree weight and proves that in complete graphs this splittability occurs with probability $C_k n^{-rac{k-1}{2}}$, while in dense random graphs it is at least $C_k n^{-rac{k}{2}-O(1)}$, thus enabling rejection-sampling approaches. It also constructs sparse, grid-like counterexamples where splittability is exponentially unlikely, implying exponential-time barriers for ReCom and related algorithms. The results establish the first dense-graph class with provably high splittability, clarifying when sampling balanced partitions is tractable and outlining open directions for understanding splittability in broader graph families.

Abstract

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.

Figures (2)

• Figure 1: Graph described in Theorem \ref{['thm:slack']} with $n=14$: (a) An almost balanced partition, with piece sizes $\{13, 14, 15\}$ and (b) an example of a balanced partition, with piece sizes $\{14, 14, 14\}$.
• Figure 2: Total spanning tree weights of all Partitions of size $i,N-i$ on the Complete Graph with $N=8$ (a) and $N=16$ (b).

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof