## New metaheuristic approaches for the edge-weighted $$k$$-cardinality tree problem.(English)Zbl 1107.90044

Summary: We propose three metaheuristic approaches, namely a tabu search, an evolutionary computation and an ant colony optimization approach, for the edge-weighted $$k$$-cardinality tree (KCT) problem. This problem is an $$NP$$-hard combinatorial optimization problem that generalizes the well-known minimum weight spanning tree problem. Given an edge-weighted graph $$G=(V,E)$$, it consists of finding a tree in $$G$$ with exactly $$k\leq |V|-1$$ edges, such that the sum of the weights is minimal. First, we show that our new metaheuristic approaches are competitive by applying them to a set of existing benchmark instances and comparing the results to two different tabu search methods from the literature. The results show that these benchmark instances are not challenging enough for our metaheuristics. Therefore, we propose a diverse set of benchmark instances that are characterized by different features such as density and variance in vertex degree. We show that the performance of our metaheuristics depends on the characteristics of the tackled instance, as well as on the cardinality. For example, for low cardinalities the ant colony optimization approach is best, whereas for high cardinalities the tabu search approach has advantages.

### MSC:

 90C59 Approximation methods and heuristics in mathematical programming 90C27 Combinatorial optimization

### Software:

K_TREE/K_SUBGRAPH; Tabu search; HC-ACO
Full Text:

### References:

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