This paper presents a new approach to efficient acoustic echo cancellation (AEC) based on reduced-rank adaptive filtering equipped with selective-decimation and adaptive interpolation. We propose a novel structure of an AEC scheme that jointly optimizes an interpolation filter, a decimation unit, and a reduced-rank filter. With a practical choice of parameters in AEC, the total computational complexity of the proposed reduced-rank scheme with the normalized least mean square (NLMS) algorithm is approximately half of that of the full-rank NLMS algorithm. We discuss the convergence properties of the proposed scheme and present a convergence condition. First, we examine the performance of the proposed scheme in a single-talk situation with an error-minimization criterion adopted in the decimation selection. Second, we investigate the potential of the proposed scheme in a double-talk situation by employing an ideal decimation selection. In addition to mean squared error (MSE) and power spectrum analysis of the echo estimation error, subjective assessments based on absolute category rating are performed, and the results demonstrate that the proposed structure provides significant improvements compared to the full-rank NLMS algorithm.