Bottleneck bichromatic plane matching of points

Given a set of n red points and n blue points in the plane, we are interested to match the red points with the blue points by straight line segments in such a way that the segments do not cross each other and the length of the longest segment is minimized. In general, this problem in NP-hard. We give exact solutions for some special cases of the input point set

On full Steiner trees in unit disk graphs

Given an edge-weighted graph G=(V,E) and a subset R of V, a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we consider the full Steiner tree problem when G is a unit disk graph. We present a 20-approximation algorithm for the full Steiner tree problem in G. As for λ-precise unit disk graphs we present a (10+1λ)-approximation algorithm, where λ is the length of the shortest edge in G

Bottleneck matchings and Hamiltonian cycles in higher-order Gabriel graphs

Given a set P of n points in the plane, the order-k Gabriel graph on P, denoted by k-GG, has an edge between two points p and q if and only if the closed disk with diameter pq contains at most k points of P, excluding p and q. It is known that 10-GG contains a Euclidean bottleneck matching of P, while 8-GG may not contain such a matching. We answer the following question in the affirmative: does 9-GG contain any Euclidean bottleneck matching of P? Thereby, we close the gap for the containment problem of Euclidean bottleneck matchings in Gabriel graphs. It is also known that 10-GG contains a Euclidean bottleneck Hamiltonian cycle of P, while 5-GG may not contain such a cycle. We improve the lower bound and show that 7-GG may not contain any Euclidean bottleneck Hamiltonian cycle of P

Approximating full steiner tree in a unit disk graph

Given an edge-weighted graph G = (V;E) and a sub- set R of V, a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we present a 20-approximation algorithm for the full Steiner tree problem when G is a unit disk graph

An optimal algorithm for the Euclidean bottleneck full Steiner tree problem

Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n+m)logm) time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model

On the hardness of full Steiner tree problems

Given a weighted graph G=(V,E) and a subset R of V, a Steiner tree in G is a tree which spans all vertices in R. The vertices in V\R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree with minimum weight. The bottleneck full Steiner tree problem is to find a full Steiner tree which minimizes the length of the longest edge. The k-bottleneck full Steiner tree problem is to find a bottleneck full Steiner tree with at most k Steiner vertices. The smallest full Steiner tree problem is to find a full Steiner tree with the minimum number of Steiner vertices. We show that the full Steiner tree problem in general graphs cannot be approximated within a factor of O(log2-ε |R|) for any ε>0. We also provide a polynomial-time approximation factor preserving reduction from the full Steiner tree problem to the group Steiner tree problem. Based o

Plane Bichromatic Trees of Low Degree

Let R and B be two disjoint sets of points in the plane such that | B| ≤ | R| , and no three points of R∪ B are collinear. We show that the geometric complete bipartite graph K(R, B) contains a non-crossing spanning tree whose maximum degree is at most max{3,⌈(|R|-1)/|B|⌉+1}; this is the best possible upper bound on the maximum degree. This proves two conjectures made by Kaneko, 1998, and solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996