On approximating tree spanners that are breadth first search trees
Abstract
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 3SAT in quasipolynomial time.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 DOI:
 10.48550/arXiv.1506.02243
 arXiv:
 arXiv:1506.02243
 Bibcode:
 2015arXiv150602243P
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 Journal of Computer and System Sciences 82 (2016) 817825