We consider the classic problem of scheduling a set of n jobs
non-preemptively on a single machine. Each job j has non-negative processing
time, weight, and deadline, and a feasible schedule needs to be consistent with
chain-like precedence constraints. The goal is to compute a feasible schedule
that minimizes the sum of penalties of late jobs. Lenstra and Rinnoy Kan
[Annals of Disc. Math., 1977] in their seminal work introduced this problem and
showed that it is strongly NP-hard, even when all processing times and weights
are 1. We study the approximability of the problem and our main result is an
O(log k)-approximation algorithm for instances with k distinct job deadlines
