Title: Equitable colouring and scheduling on identical machines

Authors: Sarah Nouri; Mourad Boudhar

Addresses: RECITS Laboratory, Faculty of Mathematics, USTHB University, BP 32 Bab-Ezzouar, El-Alia 16111, Algiers, Algeria; CERIST Center, 05 Rue des 3 frères aissou, Ben Aknoun, Algiers, Algeria ' RECITS Laboratory, Faculty of Mathematics, USTHB University, BP 32 Bab-Ezzouar, El-Alia 16111, Algiers, Algeria

Abstract: This paper deals in the first place with the problems of two-equitable colouring of a union of complete bipartite graphs and three-equitable colouring of connected bipartite graphs, where their NP-completeness is proved. In the second place, it studies the scheduling problem of conflicting jobs on identical machines, while distributing the load evenly between them. Jobs with conflicting constraints cannot be executed on the same machine, these constraints are modelled by a conflict graph. Such problem with identical processing times can be seen as an m-equitable colouring. If the conflict graph is a star graph or a union of chains, this paper demonstrates that the addressed scheduling problem remains NP-hard. Furthermore, the paper describes mixed integer linear programming formulations, followed by some heuristics. The computational experiments show that one of the MILPs can optimally solve some instances with 100 jobs, and the proposed heuristics perform well.

Keywords: equitable colouring; scheduling; conflict graphs; heuristics; mixed integer linear programming; MILP.

DOI: 10.1504/IJMOR.2023.134833

International Journal of Mathematics in Operational Research, 2023 Vol.26 No.3, pp.283 - 307

Received: 29 Jul 2022
Accepted: 02 Sep 2022
Published online: 14 Nov 2023 *

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