Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find...

We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including | but not limited to | Steiner trees for rectilinear (or iso-thetic) polygons, obstacle-avoiding Steiner trees, gro...

The rectilinear Steiner tree problem is to nd a minimum-length rectilinear interconnection of a set of points in the plane. A reduction from the rectilinear Steiner tree problem to the graph Steiner tree problem allows the use of exact algorithms for the graph Steiner tree problem to solve the rectilinear problem. Furthermore, a number of more direct, geometric algorithms have been devised for ...

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (St...

The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem; namely, the Steiner cut and the Steiner partition inequalities, as well as the a...

We obtain polynomial-time approximation-preserving reductions (up to a factor of 1+ε) from the prizecollecting Steiner tree and prize-collecting Steiner forest problems in planar graphs to the corresponding problems in graphs of bounded treewidth. We also give an exact algorithm for the prize-collecting Steiner tree problem that runs in polynomial time for graphs of bounded treewidth. This, com...

In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G = (V, E) with a cost function c : E → R and two subsets R and R satisfying R ⊂ R ⊆ V , a selected-internal Steiner tree is a Steiner tree which contains (or spans) all the vertices in R such that each vertex in R cannot be a leaf. The selected-internal Steiner tree problem is to find a selected...

In this paper, we give the first online algorithms with a polylogarithmic competitive ratio for the node-weighted prize-collecting Steiner tree and Steiner forest problems. The competitive ratios are optimal up to logarithmic factors. In fact, we give a generic technique for reducing online prize-collecting Steiner problems to the fractional version of their non-prize-collecting counterparts lo...

For any fixed parameter k ≥ 1, a tree k–spanner of a graph G is a spanning tree T in G such that the distance between every pair of vertices in T is at most k times their distance in G. In this paper, we generalize on this very restrictive concept, and introduce Steiner tree k–spanners: We are given an input graph consisting of terminals and Steiner vertices, and we are now looking for a tree k...