Given an arbitrary weighted graph with a fixed set of vertices, the Steiner tree problem is the task of finding a minimum weight subtree containing all these vertices, where the weight of a tree is the sum of the weights of the edges it contains. Steiner trees are very well studied objects in combinatorial optimization; the interest is motivated by several practical problems such as network des...

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines, In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r ∈ T is a recti...

The Steiner Tree Problem is a well know NP-complete problem that is well studied and for which fast algorithms are already available. Nonetheless, in the real world the Steiner Tree Problem is almost always accompanied by side constraints which means these approaches cannot be applied. For many problems with side constraints, only approximation algorithms are known. We introduce here a propagat...

We give a presentation of Robins and Zelikovsky’s 1.55 approximation algorithm to the Steiner Tree Problem and a thorough proof of its approximation ratio. Furthermore we sketch a proof by Thimm bounding the approximability of the Steiner Tree Problem.

This paper presents a delay-bounded minimum Steiner tree algorithm. The delay bounds, given as inputs to the algorithm, can be diierent for each individual source-sink connection. The approach is based on feasible search optimization that satisses the delay bounds rst, then improves the routing tree for the cost minimization. Iterative cut-and-link tree transformation constrained by delay bound...

This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1 − α) lnn, for a given parameter 0 < α < 1. What is the best possible running time for a...

In this paper, we consider a variation of the Euclidean Steiner Tree Problem in which the space underlying the set of nodes has a specified non-uniform cost structure. This problem is significant in many practical situations, such as laying cable networks, where the cost for laying a cable can be highly dependent on the location and the nature of the area through which it is to be laid. We empi...

We describe improvements to Smith’s branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in IR. Nodes in the B&B tree correspond to full Steiner topologies associated with a subset of the terminal nodes, and branching is accomplished by “merging” a new terminal node with each edge in the current Steiner tree. For a given topology we use a conic formulation for the problem of locat...