Approximating Euclidean Shallow-Light Trees

For a weighted graph $G = (V, E, w)$ and a designated source vertex $s \in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an $(\alpha, \beta)$-SLT of $G$ w.r.t. $s \in V$ is a spanning tree of $G$ with root-stretch $\alpha$ (preserving all distances between $s$ and the other vertices up to a factor of $\alpha$) and lightness $\beta$ (its weight is at most $\beta$ times the weight of a minimum spanning tree of $G$). Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards this question by presenting two bicriteria approximation algorithms. For any $\epsilon>0$, a set $P$ of $n$ points in constant-dimensional Euclidean space and a source $s\in P$, our first (respectively, second) algorithm returns, in $O(n \log n \cdot {\rm polylog}(1/\epsilon))$ time, a non-Steiner (resp., Steiner) tree with root-stretch $1+O(\epsilon\log \epsilon^{-1})$ and weight at most $O(\mathrm{opt}_{\epsilon}\cdot \log^2 \epsilon^{-1})$ (resp., $O(\mathrm{opt}_{\epsilon}\cdot \log \epsilon^{-1})$), where $\mathrm{opt}_{\epsilon}$ denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch $1+\epsilon$.