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This paper presents the design of two Ant Colony Optimization (ACO) approaches and their improved variants on the degree-constrained minimum spanning tree (d-MST) problem. The first approach, which we call p-ACO, uses the vertices of the construction graph as solution components, and is motivated by the well-known Prim's algorithm for constructing MST. The second approach, known as k-ACO, uses the graph edges as solution components, and is motivated by Kruskal's algorithm for the MST problem. The proposed approaches are evaluated on two different data sets containing difficult d-MST instances. Empirical results show that k-ACO performs better than p-ACO. We then enhance the k-ACO approach by incorporating the tournament selection, global update and candidate lists strategies. Empirical evaluations of the enhanced k-ACO indicate that on average, it performs better than Prufer-coded evolutionary algorithm (F-EA), problem search space (PSS), simulated annealing (SA), branch and bound (B&B), Knowles and Corne¡¦s evolutionary algorithm (K-EA) and ant-based algorithm (AB) on most problem instances from a well-known class of data set called structured hard graphs. Results also show that it is very competitive with two other evolutionary algorithm based methods, namely weight-coded evolutionary algorithm (W-EA), and edge-set representation evolutionary algorithm (S-EA) on the same class of data set.

Keywords: ant colony optimization, degree-constrained minimum spanning tree, Kruskal, metaheuristic, NP-hard, Prim, swarm intelligence

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Received August 9, 2006; revised December 6, 2006; accepted January 23, 2007.
Communicated by Chin-Teng Lin.