Abstract
We study the scheduling problem of makespan minimization with machine conflicts that arise in various settings, e.g., shared resources for pre- and post-processing of tasks or spatial restrictions. In this context, each job has a blocking time before and after its processing time, i.e., three parameters. Given a set of jobs, a set of machines, and a graph representing machine conflicts, the problem SchedulingWithMachineConflicts (smc), asks for a conflict-free schedule of minimum makespan in which the blocking times of no two jobs intersect on conflicting machines.
We show that, unless \(\text {P} =\text {NP} \), smc on m machines does not allow for a \(\mathcal {O}(m^{1-\varepsilon })\)-approximation algorithm for any \(\varepsilon >0\), even in the case of identical jobs and every choice of fixed positive parameters, including the unit case. Complementary, we provide approximation algorithms when a suitable collection of independent sets is given. Finally, we present polynomial time algorithms to solve the problem for the case of unit jobs smc-Unit on special graph classes. As our main result, we solve smc-Unit for bipartite graphs by using structural insights for conflict graphs of star forests. As the set of active machines at each point in time induces a bipartite graph, the insights yield a local optimality criterion.
D. Schmidt genannt Waldschmidt—was funded by the DFG under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Abdekhodaee, A.H., Wirth, A.: Scheduling parallel machines with a single server: some solvable cases and heuristics. Comput. Oper. Res. 29(3), 295–315 (2002)
Abdekhodaee, A.H., Wirth, A., Gan, H.S.: Equal processing and equal setup time cases of scheduling parallel machines with a single server. Comput. Oper. Res. 31(11), 1867–1889 (2004)
Abdekhodaee, A.H., Wirth, A., Gan, H.S.: Scheduling two parallel machines with a single server: the general case. Comput. Oper. Res. 33(4), 994–1009 (2006)
Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)
Baker, B.S., Coffman, E.G., Jr.: Mutual exclusion scheduling. Theoret. Comput. Sci. 162(2), 225–243 (1996)
Bodlaender, H.L., Fomin, F.V.: Equitable colorings of bounded treewidth graphs. Theoret. Comput. Sci. 349(1), 22–30 (2005)
Bodlaender, H.L., Jansen, K.: On the complexity of scheduling incompatible jobs with unit-times. In: Borzyszkowski, A.M., Sokołowski, S. (eds.) MFCS 1993. LNCS, vol. 711, pp. 291–300. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57182-5_21
Bodlaender, H.L., Jansen, K.: Restrictions of graph partition problems. Part I. Theor. Comput. Sci. 148(1), 93–109 (1995)
Bodlaender, H.L., Jansen, K., Woeginger, G.J.: Scheduling with incompatible jobs. Discret. Appl. Math. 55(3), 219–232 (1994)
Brucker, P., Dhaenens-Flipo, C., Knust, S., Kravchenko, S.A., Werner, F.: Complexity results for parallel machine problems with a single server. J. Sched. 5(6), 429–457 (2002)
Buchem, M., Kleist, L., Schmidt genannt Waldschmidt, D.: Scheduling with machine conflicts. CoRR abs/2102.08231 (2021). https://arxiv.org/abs/2102.08231
Chen, J.J., Hahn, T., Hoeksma, R., Megow, N., von der Brüggen, G.: Scheduling self-suspending tasks: new and old results. In: Proceedings of the 31st Euromicro Conference on Real-Time Systems (2019)
Chen, L., Jansen, K., Zhang, G.: On the optimality of approximation schemes for the classical scheduling problem. In: Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms, pp. 657–668 (2013)
Chrobak, M., Csirik, J., Imreh, C., Noga, J., Sgall, J., Woeginger, G.J.: The buffer minimization problem for multiprocessor scheduling with conflicts. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 862–874. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-48224-5_70
Das, S., Wiese, A.: On minimizing the makespan when some jobs cannot be assigned on the same machine. In: Proceedings of the 25th Annual European Symposium on Algorithms (2017)
Gan, H.S., Wirth, A., Abdekhodaee, A.H.: A branch-and-price algorithm for the general case of scheduling parallel machines with a single server. Comput. Oper. Res. 39(9), 2242–2247 (2012)
Gardi, F.: Mutual exclusion scheduling with interval graphs or related classes. Part I. Discret. Appl. Math. 157(1), 19–35 (2009)
Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness (1979)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45(9), 1563–1581 (1966)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)
Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)
Hall, N.G., Potts, C.N., Sriskandarajah, C.: Parallel machine scheduling with a common server. Discret. Appl. Math. 102(3), 223–243 (2000)
Hansen, P., Hertz, A., Kuplinsky, J.: Bounded vertex colorings of graphs. Discret. Math. 111(1–3), 305–312 (1993)
Håstad, J.: Clique is hard to approximate within 1- \(\varepsilon \). Acta Math. 182(1), 105–142 (1999)
Hochbaum, D.S.: Various notions of approximations: good, better, best and more. Approx. Algorithms NP-Hard Probl., 346–398 (1997)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34(1), 144–162 (1987)
Höhne, F., van Stee, R.: Buffer minimization with conflicts on a line. Theoret. Comput. Sci. 876, 25–33 (2021)
Jansen, K.: An EPTAS for scheduling jobs on uniform processors: using an MILP relaxation with a constant number of integral variables. SIAM J. Discret. Math. 24(2), 457–485 (2010)
Jansen, K., Klein, K.M., Verschae, J.: Closing the gap for makespan scheduling via sparsification techniques. Math. Oper. Res. 45(4), 1371–1392 (2020)
Jiang, Y., Zhang, Q., Hu, J., Dong, J., Ji, M.: Single-server parallel-machine scheduling with loading and unloading times. J. Comb. Optim. 30(2), 201–213 (2014). https://doi.org/10.1007/s10878-014-9727-z
Kern, W., Nawijn, W.N.: Scheduling multi-operation jobs with time lags on a single machine. In: Proceedings of the 2nd Twente Workshop on Graphs and Combinatorial Optimization (1991)
Kőnig, D.: Gráfok és mátrixok. Mat. Fizikai Lapok 38, 116–119 (1931)
Kim, M.Y., Lee, Y.H.: MIP models and hybrid algorithm for minimizing the makespan of parallel machines scheduling problem with a single server. Comput. Oper. Res. 39(11), 2457–2468 (2012)
Korte, B.H., Vygen, J.: Combinatorial Optimization, vol. 6. Springer, Heidelberg (2018)
Kravchenko, S.A., Werner, F.: Parallel machine scheduling problems with a single server. Math. Comput. Model. 26(12), 1–11 (1997)
Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: Lingas, A., Karlsson, R., Carlsson, S. (eds.) ICALP 1993. LNCS, vol. 700, pp. 40–51. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56939-1_60
Rajkumar, R., Sha, L., Lehoczky, J.P.: Real-time synchronization protocols for multiprocessors. In: Proceedings of the 9th IEEE Real-Time Systems Symposium, vol. 88, pp. 259–269 (1988)
Sahni, S.K.: Algorithms for scheduling independent tasks. J. ACM 23(1), 116–127 (1976)
Sahni, S.K.: Scheduling master-slave multiprocessor systems. IEEE Trans. Comput. 45(10), 1195–1199 (1996)
de Werra, D.: Restricted coloring models for timetabling. Discret. Math. 165, 161–170 (1997)
Xie, X., Li, Y., Zhou, H., Zheng, Y.: Scheduling parallel machines with a single server. In: Proceedings of 2012 International Conference on Measurement, Information and Control, vol. 1, pp. 453–456. IEEE (2012)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 681–690 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Buchem, M., Kleist, L., Schmidt genannt Waldschmidt, D. (2022). Scheduling with Machine Conflicts. In: Chalermsook, P., Laekhanukit, B. (eds) Approximation and Online Algorithms. WAOA 2022. Lecture Notes in Computer Science, vol 13538. Springer, Cham. https://doi.org/10.1007/978-3-031-18367-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-031-18367-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-18366-9
Online ISBN: 978-3-031-18367-6
eBook Packages: Computer ScienceComputer Science (R0)