Sampling Balanced Forests of Grids in Polynomial Time

TL;DR

This work proves that a polynomial fraction of $k$-component forests in $m\times n$ grid graphs are exactly balanced under the spanning-tree distribution, resolving a conjecture and enabling provably polynomial-time exact or approximate sampling of balanced grid partitions. The authors leverage duality and Wilson's algorithm on the dual graph to bound the probability that a uniformly random spanning tree can be split into balanced pieces, and convert this into efficient rejection-sampling procedures. They extend the results to lattice-like graphs and partitions of planar regions, establishing constant-probability multiplicative approximations and $1/\mathrm{poly}(n)$-probability additive approximations, with polynomial-time sampling algorithms in each setting. The work has practical implications for political districting analysis and other applications requiring balanced, connected partitions, linking combinatorial probability with actionable sampling methods and geometric partitioning insights.

Abstract

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.

Figures (12)

• Figure 1: A partition of a region of a lattice-like graph approximating a division of the plane given by Jordan curves, and induced by the components remaining after deleting the four bright purple edges from a spanning tree of the region. Theorem \ref{['thm:planegraph']} shows that given a division of the plane by curves, a random spanning tree of a sufficiently refined lattice-like graph can, with probability bounded below by a constant, be cut into components inducing a partition whose classes each has small Hausdorff distance from the corresponding face of the drawing.
• Figure 2: A possible run of the dual graph spanning tree sampling algorithm in the proof of Lemma \ref{['lemCentralEdgeBound']} when $m$ is odd. In this example, $m = 10$ and $n = 7$. The primal graph $G$ is depicted in gray, and the first two random walks in the dual graph $G^*$ are depicted in black.
• Figure 3: The cases in the proof of Lemma \ref{['lemCentralEdgeBound']} when $m$ is even, in which we must assume that the initial steps of the random walk from $b^*$ takes a specific path into the blue rectangle, from which it never leaves until hitting the outer face.
• Figure 4: A random walk from the bottom of the red rectangle that first exits at the very top because each of the events $\mathcal{L}_0, \mathcal{L}_{v_1,\ell}^{N0}, \mathcal{L}_{v_2,\ell}^{N0}, \dots, \mathcal{L}_{v_L,\ell}^{N0}$ occur. Here $m = 10$, $n = 16$, $\ell = 2$, and $L = 4$. The event $\mathcal{L}_0$ says that the walk from the bottom first exits the bottom purple outlined square to some vertex $v_1$ above the top. From there, each subsequent event $\mathcal{L}_{v_i, \ell}$ says that the walk exits the next green square along the top, on the side of the top boundary that is closer to the center.
• Figure 5: A possible run of the dual graph spanning tree sampling algorithm in the proof of Theorem \ref{['thm:k-split']} when $m$ is odd. In this example, $m = 12$, $n = 11$, and $k = 3$. The primal graph $G$ is depicted in gray, and the first four random walks in the dual graph $G^*$ are depicted in black.

Theorems & Definitions (55)

• Conjecture 1: Charikar, Liu, Liu, and Vuong charikar2022complexity
• Theorem 2
• Theorem 3: Informal version of Theorem \ref{['thm:planegraph']}
• Theorem 4: Informal version of Theorem \ref{['thmAdditiveError']}
• Lemma 5
• Proposition 6: Wilson
• Proposition 7
• Lemma 8
• proof : Proof that Theorem \ref{['thm:k-split']} implies Theorem \ref{['thm:k-dist']}
• Theorem 9