Scheduling of Identical Jobs with Bipartite Incompatibility Graphs on Uniform Machines. Computational Experiments
DOI:
https://doi.org/10.7494/dmms.2017.11.1-2.53Keywords:
batch scheduling, bipartite graph, polynomial algorithm, uniform machinesAbstract
In this paper, we consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard, we show that our problem can be solved to optimality in polynomial time under some restrictions imposed on the number of machines, their speeds, and the structure of the incompatibility graph. The theoretical considerations are accompanied by computer experiments with a certain model of scheduling.
References
Chen B.-L. & Yen C.-H. (2012). Equitable Δ-coloring of graphs. Discrete Mathematics, 312(9), pp. 1512–1517. DOI: https://doi.org/10.1016/j.disc.2011.05.020.
Furmańczyk H. & Kubale M. (2017). Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines. Bulletin of the Polish Academy of Sciences: Technical Sciences, 65(1), pp. 29–34. DOI: https://doi.org/10.1515/bpasts-2017-0004.
Furmańczyk H. & Kubale M. (2018). Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines. Discrete Applied Mathematics, 234, pp. 210–217. DOI: https://doi.org/10.1016/j.dam.2016.01.036.
Kosowski A. (2009). A note on the strength and minimum color sum of bipartite graphs. Discrete Applied Mathematics, 157(11), pp. 2552–2554. DOI: https://doi.org/10.1016/j.dam.2009.03.008.
Małafiejski M., Giaro K., Janczewski R. & Kubale M. (2004). Sum coloring of bipartite graphs with bounded degree. Algorithmica, 40(4), pp. 235–244. DOI: https://doi.org/10.1007/s00453-004-1111-4.
Steger A. & Wormald N.C. (1999). Generating random regular graphs quickly. Combinatorics, Probability and Computing, 8(4), pp. 377–396. DOI: https://doi.org/10.1017/S0963548399003867.
