Communication Dans Un Congrès Année : 2024

Euclidean Bottleneck Steiner Tree is Fixed-Parameter Tractable

Résumé

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)} n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach can be generalized to the problem with $\ell_p$ metric for any rational $1 \le p \le \infty$, or even other metrics on $\mathbb{R}^2$.

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hal-04778878 , version 1 (12-11-2024)

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Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xue. Euclidean Bottleneck Steiner Tree is Fixed-Parameter Tractable. SODA 2024 - 35th ACM/SIAM Symposium on Discrete Algorithms, Jan 2024, Alexandria, VA, United States. pp.699-711, ⟨10.1137/1.9781611977912.27⟩. ⟨hal-04778878⟩
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