Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Tytus Pikies; Krzysztof Turowski; Marek Kubale

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Abstract

In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. Our research stems from the works of Bodlaender, Jansen, and Woeginger (1994) and Bodlaender and Jansen (1993). In particular, we pursue the line investigated recently by Mallek, Bendraouche, and Boudhar (2019). We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ΣCj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ΣCj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.