This paper proves that the existence of a feasible schedule for a set of dependent tasks of unit execution times with release dates and deadlines on a limited number of processors is a fixed-parameter tractable problem. The parameter considered is the pathwidth of the interval graph associated with the time windows of the tasks. A fixed-parameter algorithm based on a dynamic programming approach is developed and proved to solve this decision problem. Fixed-parameter algorithms for the two classical problems P |prec, p i = 1, r i |C max and P |prec, p i = 1, r i |L max are then derived using a binary search. They are, as far as we know, the first fixed-parameter algorithms for these scheduling problems.