In this paper, a new algorithm is proposed for the solution of the Multicast Routing Problem. The algorithm is based on the Artificial Bee Colony approach hybridized with Variable Neighborhood Search. The quality of the algorithm is evaluated with experiments conducted on suitably modified benchmark instances of the Euclidean Traveling Salesman Problem from the TSP library. The results of the algorithm are compared to results obtained by several versions of the Particle Swarm Optimization algorithm. The comparisons indicated the effectiveness of the new approach.