An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem

The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degree-constrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d greater than or equal to 2, Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(\V\) time. In addition, it is shown how problem-dependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the time-complexity. Empirical results are presented for hard problem instances with up to 500 vertices, Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.