2022/06/01 by Alan Frieze, Frieze, Alan, Wesley Pegden +1 · 1 citation
Computer Science · Mathematics · #60J10 #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #cs.DS #math.CO #math.PR #msc:60J10
paper · pdf · doi:10.48550/arxiv.2206.00579
24 pages, 4 figures
arxiv created 2022/06/01 · openalex publication_date 2022/06/01 · arxiv updated 2022/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.