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[[Image:K-mst 1.pdf|thumb|An example of an undirected graph <math> G </math> with edge costs]]
[[Image:K-mst 2.pdf|thumb|The 4-MST of <math> G </math>]]
[[Image:K-mst 3.pdf|thumb|The 6-MST of <math> G </math>]]
{{DISPLAYTITLE:''k''-minimum spanning tree}}
The '''''k''-minimum spanning tree problem''', studied in [[theoretical computer science]], asks for a tree of minimum cost that has exactly ''k'' [[vertex (graph theory)|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&nbsp;''k''. The output is a tree with ''k'' vertices and ''k''&nbsp;&minus;&nbsp;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 (complexity)|reduction]] from the [[Steiner tree]] problem.
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The best approximation known for the problem achieves an approximation ratio of 2, and is by {{harvtxt|Garg|2005}}. This approximation relies heavily on the primal-dual schema of {{harvtxt|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 {{harvtxt|Arora|1998}}.
 
==References==
{{refbegin|colwidth=30em}}
*{{citation
| last = Arora | first = Sanjeev  | authorlink = Sanjeev Arora
| journal = [[Journal of the ACM]]
| title = Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
| volume = 45 | issue = 5 | pages = 753–782 | doi = 10.1145/290179.290180
| year = 1998}}.
*{{citation
| last = Garg | first = Naveen
| title = [[Symposium on Theory of Computing|Proceedings of the 37th Annual ACM Symposium on Theory of Computing]]
| contribution = Saving an epsilon: a 2-approximation for the k-MST problem in graphs
| pages = 396–402 | doi = 10.1145/1060590.1060650
| year = 2005}}.
*{{citation
| last1 = Ravi | first1 = R. | last2 = Sundaram | first2 = R. | last3 = Marathe | first3 = M. | last4 = Rosenkrantz | first4 = D. | last5 = Ravi | first5 = S.
| journal = [[SIAM Journal on Discrete Mathematics]]
| title = Spanning trees short or small
| doi = 10.1137/S0895480194266331 | volume = 9 | issue = 2 | pages = 178–200
| year = 1996}}.
*{{citation
| last1 = Goemans | first1 = M. | author1-link = Michel Goemans
| last2 = Williamson | first2 = P.
| journal = [[SIAM Journal on Computing]]
| title = A general approximation technique for constrained forest problems
| volume = 24 | issue = 2 | pages = 296–317 | doi = 10.1137/S0097539793242618
| year = 1992}}.
 
==External links==
* [http://www.nada.kth.se/~viggo/wwwcompendium/node71.html Minimum k-spanning tree in "A compendium of NP optimization problems"]
* [http://iridia.ulb.ac.be/~cblum/kctlib/ KCTLIB], KCTLIB -- A Library for the Edge-Weighted K-Cardinality Tree Problem
 
 
[[Category:Spanning tree]]
[[Category:NP-hard problems]]
 
 
{{algorithm-stub}}

Latest revision as of 17:13, 19 August 2014


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