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...

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 2...

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...

Given a surface and n fixed points on the surface, the Steiner problem asks to find the minimal length path network in the surface connecting the n fixed points. The solution to a Steiner problem is called a Steiner minimal tree. The Steiner problem in the plane has been well studied, but until recently few results have been found for non-planar surfaces. In this paper we examine the Steiner pr...

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...

We present the COST-DISTANCE problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known randomized approximation scheme for COST-DISTANCE, where is the number of sources. We reduce several common network design problems to COST-DISTANCE, obtaining (in some cases) the first ...

Selected-internal Steiner minimum tree problem is a generalization of original Steiner minimum tree problem. Given a weighted complete graph G = (V,E) with weight function c, and two subsets R ′ ( R ⊆ V with |R − R ′ | ≥ 2, selected-internal Steiner minimum tree problem is to find a Steiner minimum tree T of G spanning R such that any leaf of T does not belong to R ′ . In this paper, suppose c ...

The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems. An instance of the Covering Steiner problem consists of an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the requ...

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Stein...

We present geometric conditions that can be used to restrict or eliminate candidate topologes for Euclidean Steiner minimal trees in , d ≥ 2. Our emphasis is on conditions that are not restricted to the planar case (d = 2). For trees with a Steiner topology we give restrictions on terminal-Steiner connections that are based on the Voronoi diagram associated with the set of terminal nodes. We th...