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# Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets

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## Cite As

Anubhav Dhar, Soumita Hait, and Sudeshna Kolay. Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.25

## Abstract

The Euclidean Steiner Minimal Tree problem takes as input a set P of points in the Euclidean plane and finds the minimum length network interconnecting all the points of P. In this paper, in continuation to the works of [Du et al., 1987] and [Weng and Booth, 1995], we study Euclidean Steiner Minimal Tree when P is formed by the vertices of a pair of regular, concentric and parallel n-gons. We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for P. We also consider point sets P of size n where the number of input points not on the convex hull of P is f(n) ≤ n. We give an exact algorithm with running time 2^𝒪(f(n) log n) for such input point sets P. Note that when f(n) = 𝒪(n/(log n)), our algorithm runs in single-exponential time, and when f(n) = o(n) the running time is 2^o(n log n) which is better than the known algorithm in [Hwang et al., 1992]. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P = NP [Garey et al., 1977]. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets [Scott Provan, 1988]. In this paper, we show that if the number of input points in P not belonging to the convex hull of P is 𝒪(log n), then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any ε ∈ (0,1], when there are Ω(n^ε) points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P = NP.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Steiner minimal tree
• Euclidean Geometry
• Almost Convex point sets
• FPTAS
• strong NP-completeness

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## References

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