图中多连通分支的最优停止
Optimal stopping for many connected components in a graph
摘要：
本文研究了一个新的最优停止问题：设$G$是一个固定图，其中的$n$顶点按随机顺序在时间上一个接一个地变为在线活动的。$G$的活动部分是由活动顶点导出的子图。找到一个停止算法，使$G$的活动部分的连接组件的期望数量最大化。我们证明，如果$G$是一个$k$树，那么没有渐近更好的算法比“等到$\frac{1}{k+1}$分数的顶点”。连接组件的最大预期数量等于$$\left(\frac{k^k}{(k+1)^{k+1}}+o(1)\right)n.$$
英文摘要：
We study a new optimal stopping problem: Let $G$ be a fixed graph with $n$ vertices which become active on-line in time, one by another, in a random order. The active part of $G$ is the subgraph induced by the active vertices. Find a stopping algorithm that maximizes the expected number of connected components of the active part of $G$. We prove that if $G$ is a $k$-tree, then there is no asymptotically better algorithm than `wait until $\frac{1}{k+1}$ fraction of vertices'. The maximum expected number of connected components equals to $$\left(\frac{k^k}{(k+1)^{k+1}}+o(1)\right)n.$$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
