Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines - Hanen, Claire; Munier Kordon, Alix (Journal of scheduling, 2024)
Scheduling problems involving a set of dependent tasks with release dates and deadlines on a limited number of resources have been intensively studied. However, few parameterized complexity results exist for these problems. This paper studies the existence of a feasible schedule for a typed task system with precedence constraints and time intervals ( r i , d i ) for each job i . The problem is denoted by P | M j ( t y p e ) , p r e c , r i , d i | ⋆ . Several parameters are considered: the pathwidth pw ( I ) of the interval graph I associated with the time intervals ( r i , d i ) , the maximum processing time of a task p max and the maximum slack of a task s ℓ max . This paper establishes that the problem is para- NP -complete with respect to any of these parameters. It then provides a fixed-parameter algorithm for the problem parameterized by both parameters pw ( I ) and min ( p max , s ℓ max ) . It is based on a dynamic programming approach that builds a levelled graph which longest paths represent all the feasible solutions. Fixed-parameter algorithms for the problems P | M j ( t y p e ) , p r e c , r i , d i | C max and P | M j ( t y p e ) , p r e c , r i | L max are then derived using a binary search.