Approximate Light Spanners in Planar Graphs

TL;DR

The paper advances the design of near-optimal, light-weight planar spanners by introducing iterative planar pruning, a technique that leverages planarity to produce a $(1+\epsilon\cdot 2^{O(\log^* (1/\epsilon))})$-spanner with total weight $O(1)\cdot w(G_{OPT,\epsilon})$ in polynomial time. The approach combines a pruning framework with a laminar-structure property and dynamic programming to identify heavy edges in a current spanner that can be replaced by shorter, near-optimal paths with controlled stretch. Key contributions include a rigorous existence proof of pruning pairs, a detailed DP-based method to compute replacement paths, and a careful weight-stretch analysis that ensures exponential improvement in the weight bound through iterative rounds. The work also establishes NP-hardness for the exact minimum-weight $(1+\epsilon)$-spanner in planar graphs and provides a hard instance showing limits of the greedy spanner, underscoring the practical value of the pruning technique. Overall, the results offer a principled, planarity-aware path to constant-factor approximations for light planar spanners with favorable runtime and structural properties.

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 [...]

Figures (12)

• Figure 1: In this example, the optimal $(1+\epsilon)$-spanner contains all the red and orange edges, but a greedy $(1+\epsilon)$-spanner might contain all edges in the graph except for the red edge $(u_0, v_0)$.
• Figure 2: There could be multiple ladder structures that are hanging on a single critical path, colored red. The blue edges forming the ladders are removable edges of the current spanner $H$.
• Figure 3: An example for an edge $(a, b)$ that is $\kappa$-hanging at $(v_i, v_j)$ on path $\rho$. The orange path between $a$ and $b$ has length at most $(1+\epsilon)\cdot\mathbf{w}(a, b)$; the orange sub-path of $\rho$ between $v_i$ and $v_j$, which is also a sub-path of the orange path between $a$ and $b$, has length at least $\kappa\cdot\mathbf{w}(a, b)$.
• Figure 4: In this figure, the red path is $\rho[s,t,L]$, and $f = (x, y)$ appears many times in the multi-set $P[s, t, L]$. Then, we can show that $f$ can hang on the path $\rho[s, t, L]$ at multiple positions, say $(u_1, v_1), (u_2, v_2),\ldots ,(u_9, v_9)$. Then, we can replace the sub-path $\rho[u_1, v_9]$ with a shortcut $u_1\rightarrow (x, y)\rightarrow v_9$, which reduces the total weight of $\rho$ by at least $\mathbf{w}(f)$.
• Figure 5: In this example, $e_1 = (a_1, b_1), e_2 = (a_2, b_2)$ and region $R_{e_1}$ is contained within region $R_{e_2}$.

Theorems & Definitions (47)

• Theorem 1.2
• Theorem 1.3
• Definition 2.1: $\kappa$-hang
• Lemma 2.2: structural property, simplified
• Theorem 4.1
• proof : Proof of \ref{['thm:main']}
• Definition 4.2
• Lemma 4.3
• proof
• Lemma 4.4