We discuss an approach for solving combinatorial optimization problems using chaotic dynamics. We show the effectiveness of chaotic dynamics for solving combinatorial optimization problems by applying the chaotic neural network to quadratic assignment problems. We investigate solvable performance of the chaotic neural networks by comparing one of the conventional methods, the mutual connection neural networks. We also examine the relation between the solvable performance and Lyapunov dimensions of the chaotic neural networks. Then we show that solvable performance becomes higher when the Lyapunov dimensions take relatively smaller values, which suggests that higher solvable performance would be obtained at the edge of chaos.