@article{Pikies_Turowski_Kubale_2021, title={Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines}, volume={31}, url={https://ojs.aaai.org/index.php/ICAPS/article/view/15970}, DOI={10.1609/icaps.v31i1.15970}, abstractNote={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. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (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 ∑Cj 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.}, number={1}, journal={Proceedings of the International Conference on Automated Planning and Scheduling}, author={Pikies, Tytus and Turowski, Krzysztof and Kubale, Marek}, year={2021}, month={May}, pages={262-270} }