一种在Gibbs均匀阈值下对$G(n,d/n)$进行着色采样的简单算法
A simple algorithm for sampling colourings of $G(n,d/n)$ up to Gibbs Uniqueness Threshold
摘要：
图$G$的近似随机$k$ -着色是计算机科学和统计物理中的一个重要研究课题。它相当于构造了一个$k$ -着色的$G$，它分布接近{ \em吉布斯分布}在多项式时间。在这里，我们处理的问题时，底层图是一个实例的Erdens-Rényi随机图$G(n,d/n)$，其中$d$是一个足够大的常数。提出了一种新的近似随机$k$的有效算法--对任意$k\geq (1+ε)d$的$G(n,d/n)$着色.更具体地说，对于输入实例$G(n,d/n)$和$k\geq (1+ε)d$，该算法返回$k$着色，其分布在距输入图实例的吉布斯分布的总变化距离$n^{-Ω(1)}$内。我们提出的算法既不是MCMC算法，也不是受统计物理学家提出的消息传递算法的启发。大致的想法如下：最初，我们删除足够多的输入图的边。这导致了一个“简单的图”，它可以被有效地随机着色为$k$。算法随机地给这个简单的图着色。然后，它将一个接一个地放回删除的边。每次新的边被放回时，算法更新图的着色，使得着色保持随机。该算法的性能在很大程度上取决于吉布斯分布的某些空间相关衰减特性。
英文摘要：
Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdős-Rényi random graph $G(n,d/n)$, where $d$ is a sufficiently large constant. We propose a novel efficient algorithm for approximate random $k$-colouring $G(n,d/n)$ for any $k\geq (1+ε)d$. To be more specific, with probability at least $1-n^{-Ω(1)}$ over the input instances $G(n,d/n)$ and for $k\geq (1+ε)d$, the algorithm returns a $k$-colouring which is distributed within total variation distance $n^{-Ω(1)}$ from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a "simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution.
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