Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines

• Published in: Journal of scheduling Vol. 27; no. 2; pp. 119 - 133
• Main Authors: Hanen, Claire, Munier Kordon, Alix
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2024; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Artificial Intelligence; Business and Management; Calculus of Variations and Optimal Control; Optimization; Computer Science; Dynamic programming; Intervals; Operations Research; Operations Research/Decision Theory; Optimization; Parameterization; Precedence constraints; Scheduling; Supply Chain Management; Task scheduling; Typed task system; Precedence constraints; Fixed-parameter tractable algorithm
• ISSN: 1094-6136; 1099-1425
• DOI: 10.1007/s10951-023-00788-4

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.