As the costs for maintaining computer communication networks are rapidly rising, it is particularly important to design the network efficiently. The objective of this thesis is to model the minimum cost design of digital communications networks and propose a heuristical solution approach to the formulated model. The minimum cost design has been modeled as a zero-one integer programming problem. The Lagrangian relaxation method and subgradient optimization procedure have been used to find reasonably good feasible solutions. Although the reliability for computer communication networks is as important as the cost factor, only the cost factor is considered in the context of this thesis.