Approximate Light Spanners in Planar Graphs
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2025
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| _version_ | 1866915568707174400 |
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| author | Le, Hung Solomon, Shay Than, Cuong Tóth, Csaba D. Zhang, Tianyi |
| author_facet | Le, Hung Solomon, Shay Than, Cuong Tóth, Csaba D. Zhang, Tianyi |
| contents | 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 [...] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Light Spanners in Planar Graphs Le, Hung Solomon, Shay Than, Cuong Tóth, Csaba D. Zhang, Tianyi Data Structures and Algorithms 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 [...] |
| title | Approximate Light Spanners in Planar Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2505.24825 |