We study the Primal Dual approach in approximation algorithms for the NP hard problems of: Steiner tree problem andPrize collecting steiner tree problem.

The Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-time heuristic with an approximation ratio approaching 1 + 2 1:55, which improves upon the previously best

this problem and present an algorithm with an asymptotic approximation factor of |S|/4. This gives a sufficient condition for the existence of k edge-disjoint Steiner trees in a graph in terms of the edge-connectivity of the graph. We will show that this condition is the best possible if the number

The Euclidean Steiner tree problem is solved by finding the tree with minimal Euclidean length spanning a set of fixed vertices in the plane, while allowing for the addition of auxiliary vertices (Steiner vertices). Steiner trees are widely used to design real-world structures like highways and oil

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance

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. 1

We present a computational study of exact algorithms for the Euclidean and rectilinear Steiner tree problems in the plane. These algorithms --- which are based on the generation and concatenation of full Steiner trees --- are much more efficient than other approaches and allow exact solutions

Abstract: The Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. In this paper, we define a special class of minimum Gilbert networks, called pseudo-Gilbert–Steiner trees, and we show that it can be constructed

. In this paper, we consider preferred direction multi-layer routing, which more closely models practical applications. We adapt a well known rectilinear planar Steiner tree heuristic, resulting in a new method to construct low cost Steiner trees under a realistic model. Our implementation is fast and effective

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