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Traditional genetic algorithms often meet the occurrence of slow convergence and enclosure competition. We present a parallel algorithm for the traveling salesman problem (TSP) which incorporates several greedy heuristics based on genetic algorithms. We call this algorithm the greedy genetic algorithm (GGA) here. The ideas we incorporate in our algorithm include: (i) generating the initial population using the gene bank; (ii) double-directional greedy crossover and the local searches of greedy mutation; (Hi) special-purpose immigrations to promote diversity of population and an open competition; (iv) stepwise parallel optimization of each individual of the population; (v) developing an overall design that attempts to strike a right balance between diversification and a bias towards fitter individuals. We test all these ideas to assess their impacts on the GGA performance, and also compare our final algorithm to the simple genetic algorithm on the benchmark instances in TSPLIB, a well-known library of TSP instances. We find the greedy genetic algorithm to be a very effective and robust algorithm for TSPs.