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· O( 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( n). Specifically, let m and t be integers, such that m>0 and t≥ 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· o(( 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.