Scheduling on uniform machines with a conflict graph: complexity and resolution

• Published in: International transactions in operational research Vol. 31; no. 2; pp. 863 - 888
• Main Authors: Mallek, Amin, Boudhar, Mourad
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.03.2024
• Subjects: Completion time; Complexity; conflict graph; Graphs; incompatible jobs; Integer programming; Linear programming; Mixed integer; Operations research; Optimization; Run time (computers); Scheduling; Trees (mathematics); uniform machines; unit‐time jobs; Upper bounds
• ISSN: 0969-6016; 1475-3995
• DOI: 10.1111/itor.13170

This paper deals with the problem of scheduling a set of unit‐time jobs on a set of uniform machines. The jobs are subject to conflict constraints modeled by a graph G called the conflict graph, in which adjacent jobs cannot be processed on a same machine. The objective considered herein is the minimization of maximum job completion time in the schedule, which is famous to be NP‐hard in the strong sense. The first part of this paper is an extensive study of the computational complexity of the problem restricted to several graph classes, namely: split graphs, interval graphs, forests, trees, paths and cycles. Afterward, we focus on the resolution of the problem with arbitrary conflict graphs. For this latter, a combination of a mixed integer linear programming (MILP) formulation, lower and upper bounds is proposed. A wild range of computational experiments proved the efficiency of this technique to tremendously reduce runtime and produce more optimal solutions (around 80% in average). Furthermore, a deep analysis of the resolution process based on both the density of the conflict graph as well as machine speeds (including identical machines) is thoroughly reported.