Is spanning tree NP-complete?

Is spanning tree NP-complete?

Since removing all edges viv decrease the degree of each v by k−2, T is a 2-SPANNING-TREE of G. Thus, k-SPANNING-TREE is NP-complete for any k ≥ 2.

What is an example of an NP complete problem?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems.

How do you prove a problem is NP-complete?

We say X is NP-complete if: X ∈ NP • for all Y ∈ NP, Y ≤P X. If these hold, then X can be used to solve every problem in NP. Therefore, X is definitely at least as hard as every problem in NP.

How many NP-complete problems are there?

This list is in no way comprehensive (there are more than 3000 known NP-complete problems). Most of the problems in this list are taken from Garey and Johnson’s seminal book Computers and Intractability: A Guide to the Theory of NP-Completeness, and are here presented in the same order and organization.

Is minimum spanning tree in NP-complete or in NP hard?

The problem of finding the Steiner tree of a subset of the vertices, that is, minimum tree that spans the given subset, is known to be NP-Complete.

Is Hamiltonian path NP-complete?

Any Hamiltonian Path can be made into a Hamiltonian Circuit through a polynomial time reduction by simply adding one edge between the first and last point in the path. Therefore we have a reduction, which means that Hamiltonian Paths are in NP Hard, and therefore in NP Complete.

What is NP-hard and NP-complete problems?

A problem is NP-hard if all problems in NP are polynomial time reducible to it, even though it may not be in NP itself. If a polynomial time algorithm exists for any of these problems, all problems in NP would be polynomial time solvable. These problems are called NP-complete.

What is P NPC and NPH problems?

NP Hard Problems (NPH) In other words if a NPH problem is non-deterministic polynomial time solvable, it is a NPC problem. Example of a NPH problem that is not NPC is Halting Problem (halting problem is undecidable and all undecidable problem is guaranteed not to be in NP and hence not NPC also).

Is independent set NP-complete?

The independent set decision problem is NP-complete, and hence it is not believed that there is an efficient algorithm for solving it. The maximum independent set problem is NP-hard and it is also hard to approximate.

Is shortest path NP-complete?

Since it is also in NP, it is NP-Complete. The shortest path on the other hand is a different one, it asks what is the shortest way from point A to point B, and it is in P because there is a polynomial time algorithm that solves it (Dijkstra’s algorithm, Bellman-Ford, BFS for non weighted graphs).

Is Battleship NP-complete?

This reduction proves that BATTLESHIPS is NP-hard. It is easy to see that a non-deterministic guess is checked to be a solution in polynomial time. Hence BATTLESHIPS is NP-complete.

Why can’t a spanning tree be a leaf?

That it lies in the path to v1 from some vertex that is presently in the spanning tree. Thus, it can’t be a leaf (since leaves have no children).

What is the NP-complete problem in graph theory?

The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. Integer programming.

What are the different types of NP-complete problems?

Variants include the rural postman problem. NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. Exact cover problem. Remains NP-complete for 3-sets.

What are NP-complete subgraphs?

Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.