Approximation algorithms for solving the 1-line Euclidean minimum Steiner tree problem

Constrained Geometric Optimization Problems

“…Chen and Zhang gave an O(n 2 )time algorithm to solve this problem [40]. Similar problems have also been studied by Li et al [72] building on the research of Holby [69]. The two settings they study are: (a) the points of P lie anywhere in R 2 and must connect to the input line using any number of Steiner points, and any part of the input line used in a spanning tree does not count towards its length; and (b) the same problem, but the optimal line to minimize the network length is not given and must be computed.…”

“…Li et al provide 1.214-approximation 1 algorithms for both (a) and (b) in O(n log n) and O(n 3 log n) time respectively. Our problem differs from the problems of Chen and Zhang [40], Li et al [72], and Holby [69] in the following ways: (a) we have no restriction on the placement of the points of P with respect to the input line γ; (b) our problem does not require connecting to γ; and (c) travel in our network has the same cost on γ as off of it. For example, if the points of the set P were close to the line but far from each other, then the solution of Li et al [72] would connect the points to the line and get a tree with much lower weight/length than even the MStT.…”

“…Our problem differs from the problems of Chen and Zhang [40], Li et al [72], and Holby [69] in the following ways: (a) we have no restriction on the placement of the points of P with respect to the input line γ; (b) our problem does not require connecting to γ; and (c) travel in our network has the same cost on γ as off of it. For example, if the points of the set P were close to the line but far from each other, then the solution of Li et al [72] would connect the points to the line and get a tree with much lower weight/length than even the MStT. Such an example is shown in Figure 2.1: the MStT of points {a, b, c, d} is the same as its MST since all triples form angles larger than 2π 3 [31,58].…”

“…Such an example is shown in Figure 2.1: the MStT of points {a, b, c, d} is the same as its MST since all triples form angles larger than 2π 3 [31,58]. In our Depicted are the MST of {a, b, c, d} in red dashed line segments and its length; the input line γ; a spanning tree of {a, b, c, d} connecting each point to γ and the length of this spanning tree for the setting of Holby [69] and Li et al [72]. The numbers have been rounded to three decimal places.…”

“…setting, the MST is the best solution for this point set, whereas in the setting of Holby [69] and Li et al [72], the best solution connects each input point directly to γ to form a spanning tree between the points using pieces of γ. The length of the MST is significantly larger than the length of the other solution since in their setting, only the edges connecting the points to γ contribute to the length of the spanning tree.…”