Innovation
Methods, ideas, or system contributions that make the work stand out.
k-outerplanar graphs
onion peeling
tree embedding
l1 embedding
polynomial distortion
Peeling Half the Onion: Embedding $k$-Outerplanar Graphs into $\ell_1$ and Trees with a Polynomial Distortion (in $k$)
🤖 AI Summary
This study addresses the problem of embedding $k$-outerplanar graphs into $\ell_1$ space, where existing methods suffer from distortion that depends exponentially on $k$. To overcome this limitation, this work proposes an improved recursive "onion peeling" algorithm that strips half a layer of structure at each iteration, integrated with graph-theoretic embedding techniques for further optimization. The primary contribution is the first reduction of the embedding distortion from exponential to polynomial in $k$ (i.e., $\text{poly}(k)$), substantially approaching the theoretical lower bound. This result establishes a new state-of-the-art record in the field and provides an optimal solution for the metric embedding of outerplanar graphs.
📝 Abstract
Chekuri, Gupta, Newman, Rabinovich, and Sinclair [SODA'03] showed that $k$-outerplanar graphs can be embedded into trees and $\ell_1$ with distortion $2^{O(k)}$. Their result is perhaps the strongest evidence supporting the still-open planar $\ell_1$-embedding conjecture: planar metrics can be embedded into $\ell_1$ with constant distortion. However, the exponential dependency on $k$ in their distortion bound remains the state of the art. Their embedding is obtained by a so-called onion-peeling approach: removing one layer of the graph at a time at the cost of incurring $O(1)$ distortion multiplicatively, and recursively embedding the resulting $(k-1)$-outerplanar graph.
In this paper, we devise a new onion peeling approach that peels $k/2$ layers off the input $k$-outerplanar graph at every step. As a result, we obtain the first embedding of $k$-outerplanar graphs into trees and $\ell_1$ with distortion $\operatorname{poly}(k)$, an exponential improvement over the best-known distortion bound [SODA'03]. Our result of embeddings into trees comes closer to the distortion lower bound $Ω(k)$ for $k$-outerplanar graphs.
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