关于宽度优先搜索树的近似树空间
On approximating tree spanners that are breadth first search trees
摘要：
图$G$的树$t$-$T$是$G$的生成树，使得每对顶点之间的$T$中的距离至多是它们之间的$G$中的距离的$t$倍。当图$G$允许树$t$-树时，有一些有效的算法可以找到图$G$的树$t\cdot O(\log n)$-树.在本文中，搜索空间被缩小到$v$ -集中生成树，一个简单的家庭，包括所有的广度优先搜索树从顶点$v$。在这种情况下，不容易找到几乎在因子$o(\log n)$内的近似树空间。具体地，令$m$和$t$为整数，使得$m>0$和$t\geq 7$。如果存在一个有效的算法，它接收一个图$G$和一个顶点$v$作为输入，并返回一个$G$的$v$-集中树$t\cdot o((\log n)^{m/(m+1)})$ -树，当$G$允许一个$v$ -集中树$t$ -树时，则存在一个在准多项式时间内判定3-SAT的算法.
英文摘要：
A tree $t$-spanner $T$ of a graph $G$ is a spanning tree of $G$ such that the distance in $T$ between every pair of verices is at most $t$ times the distance in $G$ between them. There are efficient algorithms that find a tree $t\cdot O(\log n)$-spanner of a graph $G$, when $G$ admits a tree $t$-spanner. In this paper, the search space is narrowed to $v$-concentrated spanning trees, a simple family that includes all the breadth first search trees starting from vertex $v$. In this case, it is not easy to find approximate tree spanners within factor almost $o(\log n)$. Specifically, let $m$ and $t$ be integers, such that $m>0$ and $t\geq 7$. If there is an efficient algorithm that receives as input a graph $G$ and a vertex $v$ and returns a $v$-concentrated tree $t\cdot o((\log n)^{m/(m+1)})$-spanner of $G$, when $G$ admits a $v$-concentrated tree $t$-spanner, then there is an algorithm that decides 3-SAT in quasi-polynomial time.
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。
