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Degree-constrained spanning tree

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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.

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} .

References

  1. 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.
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