# Subexponential mixing for partition chains on grid-like graphs

arXiv: https://arxiv.org/abs/2206.00579 · HTML: https://mlsanity.com/paper/2206.00579/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2022.

## Built on by (most cited first, 1 of 1)

- [The Marked Edge Walk: A Novel MCMC Algorithm for Sampling of Graph Partitions](https://mlsanity.com/paper/2510.17714/) (2025)

## Abstract (arXiv)

We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of G, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bounded-degree graphs without large expander subgraphs (for example, we obtain O(2√ n) time algorithms for square grids). In the case where we instead want partitions with a small number of pieces of linear size, we show that Glauber dynamics can have exponential mixing time, even just for the case of 2 pieces, and even for 2-connected subgraphs of the grid with bounded bandwidth.