k-minimum spanning tree

An example of an undirected graph $G$ with edge costs

The 4-MST of $G$

The 6-MST of $G$

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

The input to the problem consists of an undirected graph with weights on its edges, and a number k. The output is a tree with k vertices and k − 1 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 k-MST problem has been shown to be NP-hard by a reduction from the Steiner tree problem.

The best approximation known for the problem achieves an approximation ratio of 2, and is by Garg (2005). This approximation relies heavily on the primal-dual schema of Goemans & Williamson (1992). 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 Arora (1998).

References

• Arora, Sanjeev (1998), "Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems", Journal of the ACM, 45 (5): 753–782, doi:10.1145/290179.290180.
• Garg, Naveen (2005), "Saving an epsilon: a 2-approximation for the k-MST problem in graphs", Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 396–402, doi:10.1145/1060590.1060650.
• Ravi, R.; Sundaram, R.; Marathe, M.; Rosenkrantz, D.; Ravi, S. (1996), "Spanning trees short or small", SIAM Journal on Discrete Mathematics, 9 (2): 178–200, doi:10.1137/S0895480194266331.
• Goemans, M.; Williamson, P. (1992), "A general approximation technique for constrained forest problems", SIAM Journal on Computing, 24 (2): 296–317, doi:10.1137/S0097539793242618.

External links

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