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

Abstract

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.

Variable Neighborhood Search for Minimizing the Makespan in a Uniform Parallel Machine Scheduling

“…Berndt et al [38] proposed improved support size bounds for integer linear programs (ILPs), leading to a faster approximation algorithm for makespan minimization on uniform parallel machine scheduling. Mallek and Boudhar [39] addressed scheduling on uniform parallel machines with conflict graphs. The authors proposed a mixed-integer line program (MILP) formulation, along with lower and upper bounds, to minimize the makespan.…”

Fair Allocation of Indivisible Items with Conflict Graphs

“…Also scheduling problems, where the allocation of jobs to machines is subject to pairwise conflicts between certain jobs, should be named as a related optimization problem. The resulting complexity and approximation questions were considered, e.g., in [15,32,35], and most recently in [46].…”

“…When items represent tasks with a starting and end time, each agent should be allocated a fair subset of non-overlapping tasks. Again, the mutual exclusion of two tasks/items, will be represented by the edges of a conflict graph (see, e.g., [32,46]). Note that in [20] a general treatment of conflict graphs was performed for the COIN OR Branch-and-Cut (CBC) solver.…”