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This paper presents a genetic algorithm (GA) with specialized encoding, initialization, and local search operators to optimize the design of communication network topologies. This NP-hard problem is often highly constrained so that random initialization and standard genetic operators usually generate infeasible networks. Another complication is that the fitness function involves calculating the all-terminal reliability of the network, which is a computationally expensive calculation. Therefore, it is imperative that the search balances the need to thoroughly explore the boundary between feasible and infeasible networks, along with calculating fitness on only the most promising candidate networks. The algorithm results are compared to optimum results found by branch and bound and also to GA results without local search operators on a suite of 79 test problems. This strategy of employing bounds, simple heuristic checks, and problem-specific repair and local search operators can be used on other highly constrained combinatorial applications where numerous fitness calculations are prohibitive