# Makespan Minimization on Unrelated Parallel Machines with a Few Bags

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11343)

## Abstract

Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes $$p_{i,j}$$ time units on machine $$M_i$$. In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines ($$R||C_{max}$$) where J is partitioned into b bags $$B=(B_1,\dots , B_b)$$, and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for $$R||C_{max}$$ with bags ($$R|bag|C_{max}$$). Two machines $$M_i$$ and $$M_{i'}$$ have the same machine type if $$p_{i,j}=p_{i',j}$$ for all $$j \in J$$. We give a polynomial-time approximation scheme (PTAS) for $$R|bag|C_{max}$$ with machine types where both the number of machine types and bags are constant. This result infers the existence of a PTAS for uniform parallel machines when the number of machine speeds and number of bags are both constant. Then, we present a b/2-approximation algorithm for the graph balancing problem with $$b \ge 2$$ bags; the approximation ratio is tight for $$b=3$$ unless and this algorithm solves the graph balancing problem with $$b=2$$ bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is -hard to approximate the makespan with approximation ratio less than 3/2 for both the restricted assignment and graph balancing problems with $$b=2$$ bags and $$b=3$$ bags, respectively. We also prove that makespan minimization on uniform parallel machines with $$b=2$$ bags is strongly -hard.

## Keywords

Makespan minimization Unrelated parallel machines Approximation algorithms Scheduling Bag constraints

## References

1. 1.
Asahiro, Y., Jansson, J., Miyano, E., Ono, H., Zenmyo, K.: Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. J. Comb. Optim. 22(1), 78–96 (2011)
2. 2.
Bodlaender, H., Jansen, K., Woeginger, G.: Scheduling with incompatible jobs. Discret. Appl. Math. 55(3), 219–232 (1994)
3. 3.
Chakrabarty, D., Khanna, S., Li, S.: On (1, $$\varepsilon$$)-restricted assignment makespan minimization. In: 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1087–1101 (2015)Google Scholar
4. 4.
Das, S., Wiese, A.: On minimizing the makespan when some jobs cannot be assigned on the same machine. In: 24th Annual European Symposium on Algorithms, LIPIcs, vol. 87, pp. 31:1–31:14 (2017)Google Scholar
5. 5.
Dokka, T., Kouvela, A., Spieksma, F.: Approximating the multi-level bottleneck assignment problem. Oper. Res. Lett. 40, 282–286 (2012)
6. 6.
Ebenlendr, T., Krčál, M., Sgall, J.: Graph balancing: a special case of scheduling unrelated parallel machines. In: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 483–490 (2008)Google Scholar
7. 7.
Eisenbrand, F.: Solving an avionics real-time scheduling problem by advanced IP-methods. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 11–22. Springer, Heidelberg (2010).
8. 8.
Gairing, M., Monien, B., Woclaw, A.: A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theor. Comput. Sci. 380(1), 87–99 (2007)
9. 9.
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness (1979)Google Scholar
10. 10.
Gehrke, J., Jansen, K., Kraft, S., Schikowski, J.: A PTAS for scheduling unrelated machines of few different types. Int. J. Found. Comput. Sci. 29, 591–621 (2018)
11. 11.
Graham, R., Lawler, E., Lenstra, J., Rinnooy, K.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)
12. 12.
Huang, C., Ott, S.: A combinatorial approximation algorithm for graph balancing with light hyper edges. In: 24th Annual European Symposium on Algorithms, LIPIcs, vol. 57, pp. 49:1–49:15 (2016)Google Scholar
13. 13.
Jansen, K., Maack, M.: An EPTAS for scheduling on unrelated machines of few different types. Algorithms and Data Structures. LNCS, vol. 10389, pp. 497–508. Springer, Cham (2017).
14. 14.
Jansen, K., Rohwedder, L.: A quasi-polynomial approximation for the restricted assignment problem. In: Eisenbrand, F., Koenemann, J. (eds.) IPCO 2017. LNCS, vol. 10328, pp. 305–316. Springer, Cham (2017).
15. 15.
Kolliopoulos, S., Moysoglou, Y.: The 2-valued case of makespan minimization with assignment constraints. Inf. Process. Lett. 113(1), 39–43 (2013)
16. 16.
Lenstra, J., Shmoys, D., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1–3), 259–271 (1990)
17. 17.
Lin, Y., Li, W.: Parallel machine scheduling of machine-dependent jobs with unit-length. Eur. J. Oper. Res. 156(1), 261–266 (2004)
18. 18.
Page, D., Solis-Oba, R.: A 3/2-approximation algorithm for the graph balancing problem with two weights. Algorithms 9(2), 38 (2016)
19. 19.
Shchepin, E., Vakhania, N.: An optimal rounding gives a better approximation for scheduling unrelated machines. Oper. Res. Lett. 33, 127–133 (2005)

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## Authors and Affiliations

1. 1.Department of Computer ScienceWestern UniversityLondonCanada