The scheduling problem of minimizing total tardiness on a single machine is known to be NP-hard in the ordinary sense. In this paper, we consider the special case of the problem when the processing times pj and the due dates dj of the jobs , are oppositely ordered: p1≥p2≥⋯≥pn and d1≤d2≤⋯≤dn. It is shown that already this special case is NP-hard in the ordinary sense, too. The set of jobs N is partitioned into k,1≤k≤n, subsets M1,M2,…,Mk, Mν⋂Mμ=0̸ for ν≠μ,N=M1⋃M2⋃⋯⋃Mk, such that maxi,j∈Mν|di−dj|≤minj∈Mνpj for each ν=1,2,…,k. We propose algorithms which solve the problem: in O(kn∑pj) time if 1≤k