Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Abstract

In this paper we consider the problem of 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 work of Bodlaender et al. [1], [2]. In particular, we pursue the line investigated partially by Mallek et al. [3], where the graph is complete multipartite so each machine can do jobs only from one partition.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 easy problems, or algorithms with a guaranteed constant worst-case approximation ratio or even in some cases a PTAS for the harder ones.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.

Scheduling on uniform and unrelated machines with bipartite incompatibility graphs

“…an encoding where only the number of jobs in each part is given as input data. Recently, in [24] results for complete multipartite incompatibility graphs were presented, bringing some exact or approximate algorithms for uniform machines, and inapproximability results for unrelated ones. For example, a polynomial time algorithm for the Q|G = complete bipartite, p j = 1|C max under the customary unary encoding was given.…”