We address the problem of computing these representative cycles, termed as persistent 1-cycles, for H₁-persistent homology with Z₂ coefficients. We propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software, presented in this webpage which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.