Is there available polynomial time solution for the minimum Steiner tree problem?
Since Steiner Tree problem is NP-Hard, there are no polynomial time solutions that always give optimal cost.
Is Steiner tree NP complete?
Theorem: Steiner tree problem in graphs is NP-complete. The proof will follow step by step the template advised by Garey and Johnson [1] to show that a problem Π is NP-complete: 1.
What do you mean by rectilinear Steiner tree?
(definition) Definition: A minimum-length rectilinear tree connecting a set of points, called terminals, in the plane. This tree may include points other than the terminals, which are called Steiner points.
Can NP-complete problems be solved in polynomial time?
If a problem in NP cannot be solved in polynomial time then all problems in NP-complete cannot be solved in polynomial time. Note that an NP-complete problem is one of those hardest problems in NP.
Is Steiner tree an NP-hard problem?
In fact, the Steiner tree problem is known to be NP-hard [164], even for points in the Euclidean plane. The Steiner tree problem is in sharp contrast with the MST problem, which can be solved exactly in low-degree polynomial time.
Can mst be cyclic?
Minimum Cost Edge: If the minimum cost edge of a graph is unique, then this edge is included in any MST. For example, in the above figure, the edge AB (of the least weight) is always included in MST. If a new edge is added to the spanning tree then it will become cyclic because every spanning tree is minimally acyclic.
What is P and NP problem?
The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly.
Is the Steiner tree problem NP-hard?
In fact, the Steiner tree problem is known to be NP-hard [164 ], even for points in the Euclidean plane. The Steiner tree problem is in sharp contrast with the MST problem, which can be solved exactly in low-degree polynomial time.
What is the difference between spanning tree and Steiner tree?
Spanning Tree vs Steiner Tree. Minimum Spanning Tree is a minimum weight tree that spans through all vertices. If given subset (or terminal) vertices is equal to set of all vertices in Steiner Tree problem, then the problem becomes Minimum Spanning Tree problem.
Where is the Steiner point on a Steiner tree?
Steiner tree for three points A, B, and C (note there are no direct connections between A, B, C ). The Steiner point S is located at the Fermat point of the triangle ABC. The Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization.
Is there a polynomial-time algorithm for a Steiner tree?
There is a polynomial-time algorithm that approximates the minimum Steiner tree to within a factor of is NP-hard. For the restricted case of Steiner Tree problem with distances 1 and 2, a 1.25-approximation algorithm is known. Karpinski and Alexander Zelikovsky constructed PTAS for the dense instances of Steiner Tree problems.