On approximating tree spanners that are breadth first search trees

TL;DR

This paper investigates the difficulty of approximating tree spanners within certain factors, showing that achieving better approximations in restricted cases implies unlikely breakthroughs in computational complexity.

Contribution

It establishes hardness results for approximating $v$-concentrated tree spanners, linking improved algorithms to solving 3-SAT in quasi-polynomial time.

Findings

Efficient algorithms for better approximations would imply breakthroughs in complexity theory.

Hardness results for approximating $v$-concentrated tree spanners within certain factors.

Connections between graph spanner approximations and the complexity of 3-SAT.

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 3-SAT in quasi-polynomial time.