论文标题
较少的颜色以完美模拟正确的颜色
Fewer colors for perfect simulation of proper colorings
论文作者
论文摘要
给定图形$ g $和颜色集$ \ {1，\ ldots，k \} $，$ \ textit {正确的coloring} $是$ g $的每个顶点的颜色分配，因此没有两个边缘连接的两个顶点给定颜色相同。精心研究了从一组正确的颜色中均匀地绘制适当着色的问题。最近，Bhandari和Chakraborty开发了一种多项式时间随机算法，用于在$ k>3δ$时获得此类抽奖，其中$δ$是该图的最大程度。他们的方法将边界链与过去协议的耦合一起使用。在这里，根据作者引入的随机性回收协议，并在focs 2000填充了新的随机算法。鉴于$ n $顶点，当$ o（n \ ln（n））$ o（n \ ln（n））$ k> 2.27（δ-1）$的所有$δ\ geq egeq feeq 2 $时。
Given a graph $G$ and color set $\{1, \ldots, k\}$, a $\textit{proper coloring}$ is an assignment of a color to each vertex of $G$ such that no two vertices connected by an edge are given the same color. The problem of drawing a proper coloring exactly uniformly from the set of proper colorings is well-studied. Most recently, Bhandari and Chakraborty developed a polynomial expected time randomized algorithm for obtaining such draws when $k > 3Δ$, where $Δ$ is the maximum degree of the graph. Their approach used a bounding chain together with the coupling from the past protocol. Here a new randomized algorithm is presented based upon the randomness recycler protocol introduced by the author and Fill at FOCS 2000. Given $n$ vertices, this method takes $O(n \ln (n))$ expected steps when $k > 2.27(Δ- 1)$ for all $Δ\geq 2$.
