Subexponential mixing for partition chains on grid-like graphs

TL;DR

This paper develops subexponential algorithms for generating uniform partitions of grid-like graphs with connected pieces, using Glauber dynamics, and analyzes their mixing times.

Contribution

It introduces fixed-parameter tractable algorithms for certain partition types and demonstrates exponential mixing times in other cases, advancing understanding of sampling complexity.

Findings

Polynomial-time algorithms for partitions with linearly many small pieces on grid graphs.

Subexponential algorithms for bounded-degree graphs without large expanders.

Exponential mixing times for partitions with few large pieces, even in simple grid cases.

Abstract

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 $O(2^{\sqrt n})$ 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.