thumb|An example of an undirected graph <math> G </math> with edge costs

thumb|The 4-MST of <math> G </math>

thumb|The 6-MST of <math> G </math>

The -minimum spanning tree problem, studied in theoretical computer science, asks for a tree of minimum cost that has exactly vertices and forms a subgraph of a larger graph. It is also called the -MST or edge-weighted -cardinality tree. Finding this tree is NP-hard, but it can be approximated to within a constant approximation ratio in polynomial time.

Problem statement

The input to the problem consists of an undirected graph with weights on its edges, and a The output is a tree with vertices and edges, with all of the edges of the output tree belonging to the input graph. The cost of the output is the sum of the weights of its edges, and the goal is to find the tree that has minimum cost. The problem was formulated by and by .

Ravi et al. also considered a geometric version of the problem, which can be seen as a special case of the graph problem.

In the geometric -minimum spanning tree problem, the input is a set of points in the plane. Again, the output should be a tree with of the points as its vertices, minimizing the total Euclidean length of its edges. That is, it is a graph -minimum spanning tree on a complete graph with Euclidean distances as weights.

Computational complexity

When is a fixed constant, the -minimum spanning tree problem can be solved in polynomial time by a brute-force search algorithm that tries all -tuples of vertices.

However, for variable , the -minimum spanning tree problem has been shown to be NP-hard by a reduction from the Steiner tree problem. the same is true for the -minimum spanning tree problem.

The best approximation known for the general problem achieves an approximation ratio of 2, and is by . This approximation relies heavily on the primal-dual schema of .

When the input consists of points in the Euclidean plane (any two of which can be connected in the tree with cost equal to their distance) there exists a polynomial time approximation scheme devised by .

References

  • Minimum k-spanning tree in "A compendium of NP optimization problems"
  • KCTLIB, KCTLIB -- A Library for the Edge-Weighted K-Cardinality Tree Problem