Combining Ant Colony Optimization with Dynamic Programming for Solving the k-Cardinality Tree Problem

Research efforts in metaheuristics have shown that an intelligent incorporation of more classical optimization techniques in metaheuristics can be very beneficial. In this paper, we combine the metaheuristic ant colony optimization with dynamic programming for the application to the NP-hard k-cardinality tree problem. Given an undirected graph G with node and/or edge weights, the problem consists of finding a tree in G with exactly k edges such that the sum of the weights is minimal. In a standard ant colony optimization algorithm, ants construct trees with exactly k edges. In our algorithm, ants may construct trees that have more than k edges, in which case we use a recent dynamic programming algorithm to find—in polynomial time—the best k-cardinality tree embedded in the bigger tree constructed by the ants. We show that our hybrid algorithm improves over the standard ant colony optimization algorithm and, for node-weighted grid graph instances, is a current state-of-the-art method.