Flowshop scheduling deals with processing a set of jobs through a
set of machines, where all jobs have to pass among machines in the
same order. With the exception of minimizing a makespan on two
machines, almost all other flowshop problems in a general setup are
known to be computationally intractable. In this paper we study a
special case of flowshop defined by additional machine dominance
constraints. The main result of this paper shows that the flowshop
problem with a dominant machine (where dominance can be defined
in several ways) can be solved in a polynomial time for a broad
class of objective functions. This result is achieved by providing
a general recipe how to obtain polynomial time optimization
algorithms for particular objective functions from the corresponding
algorithms for single machine problems.