An exact dynamic programming algorithm(EDPA) is proposed for a kind of green single-machine scheduling problems, i.e., the low-carbon single-machine scheduling problem with release times and due dates. The first and second optimization objectives are the maximum tardiness and the total carbon emissions, respectively. Firstly, the permutation-based model of the considered problem is built. This model can be described by triplet$1|agr(r_j,d_j)|TCE/T_{\max