The generalized spanning tree or group Steiner problem (GSP) is a generalization of the Steiner problem in graphs (SPG): one requires a tree spanning (at least) one vertex of each subset, given in a family of vertex subsets, while minimizing the sum of the corresponding edge costs. Specialized solution procedures have been developed for this problem. In this paper we investigate the performance...

Let T be a distinguished subset of vertices in a graph G. A T Steiner tree is a subgraph of G that is a tree and that spans T . Kriesell conjectured that G contains k pairwise edge-disjoint T -Steiner trees provided that every edge-cut of G that separates T has size ≥ 2k. When T = V (G) a T -Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Wil...

In this paper, we present an algorithm called FOARS for obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) construction. FOARS applies a top-down approach which first partitions the set of pins into several subsets uncluttered by obstacles. Then an obstacle-avoiding Steiner tree is generated for each subset by an obstacle aware version of the rectilinear Steiner minimal tree (RSMT) alg...

The smallest tree that contains all vertices of a subset W of V (G) is called a Steiner tree. The number of edges of such a tree is the Steiner distance of W and union of all Steiner trees of W form a Steiner interval. Both of them are described for the lexicographic product in the present work. We also give a complete answer for the following invariants with respect to the Steiner convexity: t...

The Connected Facility Location (ConFL) problem combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facili...

We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterizati...

We discuss a new approach to constructing the rectilinear Steiner tree (RST) of a given set of points in the plane, starting from a minimum spanning tree (MST). The main idea in our approach is to find layouts for the edges of the MST, so as to maximize the overlaps between the layouts, thus minimizing,the cost (i.e., wire length) of the resulting rectilinear Steiner tree. We describe two algor...

We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut re...

Given a directed graph G and a list (s1, t1), . . . , (sk, tk) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, . . . , tk) is known to be fixed-parameter tractable parameterized by the number of term...

Depending on different switching technologies, the multicast communication problem has been formulated as three different graph theoretical problems: the Steiner tree problem, the multicast tree problem, and the multicast path problem. Our efforts in this paper are to reduce the communication traffic of multicast in hypercube multiprocessors. We propose three heuristic algorithms for the three ...