An Exact Algorithm for the Line-Constrained Bottleneck k-Steiner Tree Problem

Abstract

In this paper, we address the line-constrained bottleneck k-Steiner tree (LcBkStT) problem. Specifically, given an input line l, a set P of n points in $\mathbb {R}^2$ and a positive integer k, we are asked to find at most k Steiner points located on this line l and additionally a spanning tree $T_l$ on these $n+k$ points, the objective is to minimize the length of the longest edge in $T_l$, where the edges in $T_l$ are not allowed to cross this line l and the length of each edge in $T_l$ is equal 0 if the two endpoints of that edge are located on the aforementioned line l. Using a technique of oriented Voronoi diagram, we design an exact algorithm for the LcBkStT problem in $O(n \log n + f(k)\cdot n^k)$ time, where f(k) is a function dependent only on the positive integer k. This algorithm is an exact algorithm for the LcB1StT problem (for $k=1$) in $O(n \log n)$ time.

This paper is supported by the National Natural Science Foundation of China [Nos. 12361066, 12101593]. Junran Lichen is also supported by Fundamental Research Funds for the Central Universities [No.buctrc202219], Suding Liu is supported by the China Scholarship Council [No. 202107030013], and Jianping Li is also supported by Project of Yunling Scholars Training of Yunnan Province [No. K264202011820].