Approximate Light Spanners in Planar Graphs

Abstract

In their seminal paper, Althöfer et al. (DCG 1993) introduced the {\em greedy spanner} and showed that, for any weighted planar graph $G$, the weight of the greedy $(1+ε)$-spanner is at most $(1+\frac{2}ε) \cdot w(MST(G))$, where $w(MST(G))$ is the weight of a minimum spanning tree $MST(G)$ of $G$. This bound is optimal in an {\em existential sense}: there exist planar graphs $G$ for which any $(1+ε)$-spanner has a weight of at least $(1+\frac{2}ε) \cdot w(MST(G))$. However, as an {\em approximation algorithm}, even for a {\em bicriteria} approximation, the weight approximation factor of the greedy spanner is essentially as large as the existential bound: There exist planar graphs $G$ for which the greedy $(1+x ε)$-spanner (for any $1\leq x = O(ε^{-1/2})$) has a weight of $Ω(\frac{1}{ε\cdot x^2})\cdot w(G_{OPT, ε})$, where $G_{OPT, ε}$ is a $(1+ε)$-spanner of $G$ of minimum weight. Despite the flurry of works over the past three decades on approximation algorithms for spanners as well as on light(-weight) spanners, there is still no (possibly bicriteria) approximation algorithm for light spanners in weighted planar graphs that outperforms the existential bound. As our main contribution, we present a polynomial time algorithm for constructing, in any weighted planar graph $G$, a $(1+ε\cdot 2^{O(\log^* 1/ε)})$-spanner for $G$ of total weight $O(1)\cdot w(G_{OPT, ε})$. To achieve this result, we develop a new technique, which we refer to as {\em iterative planar pruning}. It iteratively modifies a spanner [...]