  In this paper we study the classical single machine scheduling problem where
the objective is to minimize the total weight of tardy jobs. Our analysis
focuses on the case where one or more of three natural parameters is either
constant or is taken as a parameter in the sense of parameterized complexity.
These three parameters are the number of different due dates, processing times,
and weights in our set of input jobs. We show that the problem belongs to the
class of fixed parameter tractable (FPT)problems when combining any two of
these three parameters. We also show that the problem is polynomial-time
solvable when the latter two parameters are constant, complementing Karp's
result who showed that the problem is NP-hard already for a single due date.
