论文标题
在类似网格图上的分区链的次指数混合
Subexponential mixing for partition chains on grid-like graphs
论文作者
论文摘要
我们考虑了生成图表集的均匀随机分区的问题，以使每个作品都诱导连接的子图。对于我们想要具有线性多个有界尺寸的分区的情况，我们基于Glauber动力学获得了近似的采样算法，这些算法相对于$ g $的带宽可以固定参数，并且具有简单的表达性依赖于带宽。例如，对于恒定或对数宽度的矩形，这给出了多项式时间采样算法。更一般地，这给出了无大扩展子图的有限度图的子指数算法（例如，我们获得$ O（2^{\ sqrt n}）$ time Algorithms for Square Grids）。 在我们需要使用少量线性尺寸的隔板的情况下，我们表明Glauber Dynamics可以具有指数的混合时间，甚至仅适用于2件，甚至对于带有有界带宽的2个连接的网格子图。
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.
