Steiner tree problem

The Steiner tree problem has also been investigated in higher dimensions and on various surfaces.

The Steiner tree problem has applications in circuit layout or network design.

This variant is known as the metric Steiner tree problem.

When the flow amounts are all equal, this reduces to the classical Steiner tree problem.

The Fermat point gives a solution to the geometric median and Steiner tree problems for three points.

A minimum spanning tree is a feasible but not usually optimal solution to the Steiner tree problem.

In the metric Steiner tree problems, the Steiner ratio is 2.

Most versions of the Steiner tree problem are NP-complete.

Phylomurka (Solver for the Steiner tree problem in networks)

Gilbert did important work on the Steiner tree problem in 1968, formulating it in a way that unified it with network flow problems.

In the Euclidean Steiner tree problem, the Steiner ratio is conjectured to be .

A minimum single-trunk Steiner tree problem (MSTST) may be found in linear time.

Therefore an algorithm that finds a minimum spanning tree is a polynomial-time factor-2 approximation algorithm for the metric Steiner tree problem.

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.

This concept was introduced by Rajagopalan and Vazirani who used it to provide a (3/2 + ε) approximation algorithm for the Steiner tree problem on such instances.

The k-MST problem is shown to be NP-hard by reducing the Steiner tree problem to the '-MST problem.

In a special case of the graph problem, the Steiner tree problem for quasi-bipartite graphs, S is required to include at least one endpoint of every edge in G.

For general N, the Euclidean Steiner tree problem is NP-hard, and hence it is not known whether an optimal solution can be found by using a polynomial-time algorithm.

Given an instance of the (non-metric) Steiner tree problem, we can transform it in polynomial time into an equivalent instance of the metric Steiner tree problem; the transformation preserves the approximation factor.

The Steiner tree problem is superficially similar to the minimum spanning tree problem: given a set V of points (vertices), interconnect them by a network (graph) of shortest length, where the length is the sum of the lengths of all edges.

While the Euclidean version admits a PTAS, it is known that the metric Steiner tree problem is APX-complete, i.e., it is believed that arbitrarily good approximation ratios cannot in general be achieved in polynomial time.

The minimum rectilinear Steiner tree problem (MRST) is a variant of the geometric Steiner tree problem in the plane, in which the Euclidean distance is replaced with the rectilinear distance.

The simplest routing problem, called the Steiner tree problem, of finding the shortest route for one net in one layer with no obstacles and no design rules is NP-hard if all angles are allowed and NP-complete if only horizontal and vertical wires are allowed.

Such problems arise in approximation algorithms; a famous example is the directed Steiner tree problem, for which there is a quasi-polynomial time approximation algorithm achieving an approximation factor of (n being the number of vertices), but showing the existence of such a polynomial time algorithm is an open problem.

The difference between the Steiner tree problem and the minimum spanning tree problem is that, in the Steiner tree problem, extra intermediate vertices and edges may be added to the graph in order to reduce the length of the spanning tree.