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

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 [5] and [15], 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(_logⁿ_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 [9]. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P = NP [6]. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets [14]. 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.