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We propose a positively self-feedbacked Hopfield neural network architecture for efficiently solving crossbar switch problem. A binary Hopfield neural network architecture with additional positive self-feedbacks and its collective computational properties are studied. It is proved theoretically and confirmed by simulating the randomly generated Hopfield neural network with positive self-feedbacks that the emergent collective properties of the original Hopfield neural network also are present in this network architecture. The network architecture is applied to crossbar switching and results of computer simulations are presented and used to illustrate the computation power of the network architecture. The simulation results show that the Hopfield neural network architecture with positive self-feedbacks is much better than the previous works including the original Hopfield neural network architecture, Troudet's architecture and maximum neural network for crossbar switching in terms of both the computation time and the solution quality.