Given an undirected graph with weights associated with its edges, the Steiner tree problem consists of finding a minimum-weighted subgraph spanning a given subset of nodes (terminals) of the original graph. In this paper, we describe a tabu search algorithm for the Steiner problem in graphs, based on a neighborhood defined by insertions and eliminations of Steiner nodes. Move estimations, elimination tests, and neighborhoodreduction techniques are used to speed up the local search, leading to a very fast algorithm with very good results in terms of solution quality. Computational experiments on benchmark problems are reported, comparing the behavior of the proposed tabu search algorithm with that of other heuristics from the literature. Tabu search clearly outperforms other heuristics in terms of computation times, obtaining better or comparable solutions.

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We propose a GRASP using an hybrid heuristic-subproblem optimization approach for the Multi-Level Capacitated Minimum Spanning Tree (MLCMST) problem. The motivation behind such approach is that to evaluate moves rearranging the configuration of a subset of nodes may require to solve a smaller-sized MLCMST instance. We thus use heuristic rules to define, in both the construction and the local search phases, subproblems which are in turn solved exactly by employing an integer programming model. We report numerical results obtained on benchmark instances from the literature, showing the approach to be competitive in terms of solution quality. The proposed GRASP have in fact improved the best known upper bounds for almost all of the considered instances.