On the parametric complexity of schedules to minimize tardy tasks

Given a set T of tasks, each of unit length and having an individual deadline d(t)∈ Z, a set of precedence constraints on T , and a positive integer k6 |T |, we can ask “Is there a one-processor schedule for T that obeys the precedence constraints and contains no more than k late tasks?” This is a well-known NP-complete problem. We might also inquire “Is there a one-processor schedule for T that obeys the precedence constraints and contains at least k tasks that are on time i.e. no more than |T | − k late tasks?” Within the framework of classical complexity theory, these two questions are merely di9erent instances of the same problem. Within the recently developed framework of parameterized complexity theory, however, they give rise to two separate problems that may be studied independently of one another. We investigate these problems from the parameterized point of view. We show that, in the general case, both these problems are hard for the parameterized complexity class W [1]. In contrast, in the case where the set of precedence constraints can be modelled by a partial order of bounded width, we show that both these problems are %xed parameter tractable. c © 2002 Elsevier Science B.V. All rights reserved.