In graph theory, a degree-constrained spanning tree is a spanning tree where the maximum vertex degree is limited to a certain constant k. The degree-constrained spanning tree problem is to determine whether a particular graph has such a spanning tree for a particular k.
Formal definition
Input: n-node undirected graph G(V,E); positive integer k < n.
Question: Does G have a spanning tree in which no node has degree greater than k?
NP-completeness
This problem is NP-complete (Garey & Johnson 1979). This can be shown by a reduction from the Hamiltonian path problem. It remains NP-complete even if k is fixed to a value ≥ 2. If the problem is defined as the degree must be ≤ k, the k = 2 case of degree-confined spanning tree is the Hamiltonian path problem.
Degree-constrained minimum spanning tree
On a weighted graph, a Degree-constrained minimum spanning tree (DCMST) is a degree-constrained spanning tree in which the sum of its edges has the minimum possible sum. Finding a DCMST is an NP-Hard problem.1
Heuristic algorithms that can solve the problem in polynomial time have been proposed, including Genetic and Ant-Based Algorithms.
Approximation Algorithm
Fürer & Raghavachari (1994) give an iterative polynomial time algorithm which, given a graph G {\displaystyle G} , returns a spanning tree with maximum degree no larger than Δ ∗ + 1 {\displaystyle \Delta ^{*}+1} , where Δ ∗ {\displaystyle \Delta ^{*}} is the minimum possible maximum degree over all spanning trees. Thus, if k = Δ ∗ {\displaystyle k=\Delta ^{*}} , such an algorithm will either return a spanning tree of maximum degree k {\displaystyle k} or k + 1 {\displaystyle k+1} .
- Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, ISBN 978-0-7167-1045-5. A2.1: ND1, p. 206.{{citation}}: CS1 maint: postscript (link)
- Fürer, Martin; Raghavachari, Balaji (1994), "Approximating the minimum-degree Steiner tree to within one of optimal", Journal of Algorithms, 17 (3): 409–423, CiteSeerX 10.1.1.136.1089, doi:10.1006/jagm.1994.1042.
References
Bui, T. N. and Zrncic, C. M. 2006. An ant-based algorithm for finding degree-constrained minimum spanning tree. In GECCO ’06: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 11–18, New York, NY, USA. ACM. http://www.cs.york.ac.uk/rts/docs/GECCO_2006/docs/p11.pdf ↩