This paper focuses on the use of different memory strategies to improve multistart methods. A network design problem in which the costs are given by discrete stepwise increasing cost functions of the capacities installed in the edges is used to illustrate the contributions of adaptive memory and vocabulary building strategies. Heuristics based on shortest path and maximum flow algorithms are combined with adaptive memory in order to obtain an approximate solution to the problem in the framework of a multistart algorithm. Furthermore, a vocabulary building intensification mechanism supported by the resolution of a linear program is also explored. Numerical experiments have shown that the proposed algorithm obtained the best known solutions for some instances in the literature. These results show the contribution of each memory component and the effectiveness of their combination.