# Approximation algorithms for the graph balancing problem with two speeds and two job lengths

## Abstract

Consider a set of *n* jobs and *m* uniform parallel machines, where each job has a length \(p_j \in {\mathbb {Q}}^+\) and each machine has a speed \(s_i \in {\mathbb {Q}}^+\). The goal of the graph balancing problem with speeds is to schedule each job *j* non-preemptively on one of a subset \({\mathcal {M}}_j\) of at most 2 machines so that the makespan is minimized. This is a \(\textsf {NP}\)-hard special case of the makespan minimization problem on unrelated parallel machines, where for the latter a longstanding open problem is to find an approximation algorithm with approximation ratio better than 2. Our main contribution is an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio \((\sqrt{65}+7)/8 \approx 1.88278\). In addition, we consider when every \({\mathcal {M}}_j\) has no cardinality constraints, this is the restricted assignment problem in the uniform parallel machine setting. We present a simple \((2-\alpha /\beta )\)-approximation algorithm in this case when every job has one of two job lengths \(p_j \in \{\alpha , \beta \}\) where \(\alpha < \beta \).

## Keywords

Makespan minimization Unrelated parallel machines Approximation algorithms Graph balancing problem Restricted assignment problem Scheduling theory## Mathematics Subject Classification

68W25 90B35## Notes

### Acknowledgements

We would like to thank Marten Maack for his comments on our paper. In addition, we would like to acknowledge and thank the anonymous reviewers for their criticisms.

## References

- Asahiro Y, Jansson J, Miyano E, Ono H, Zenmyo K (2011) Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. J Comb Optim 22(1):78–96MathSciNetCrossRefzbMATHGoogle Scholar
- Chakrabarty D, Shiragur K (2016) Graph balancing with two edge types. arXiv preprint arXiv:1604.06918
- Chakrabarty D, Khanna S, Li S (2015) On (1, \(\varepsilon \))-restricted assignment makespan minimization. In: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 1087–1101Google Scholar
- Ebenlendr T, Krčál M, Sgall J (2008) Graph balancing: a special case of scheduling unrelated parallel machines. In: Proceedings of the nineteenth annual ACM–SIAM symposium on discrete algorithms (SODA), pp 483–490Google Scholar
- Ebenlendr T, Krčál M, Sgall J (2014) Graph balancing: a special case of scheduling unrelated parallel machines. Algorithmica 68(1):62–80MathSciNetCrossRefzbMATHGoogle Scholar
- Gairing M, Monien B, Woclaw A (2007) A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theor Comput Sci 380(1):87–99MathSciNetCrossRefzbMATHGoogle Scholar
- Graham R, Lawler E, Lenstra J, Rinnooy K (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discret Math 5:287–326MathSciNetCrossRefzbMATHGoogle Scholar
- Huang C, Ott S (2016) A combinatorial approximation algorithm for graph balancing with light hyper edges. In: 24th Annual European symposium on algorithms (ESA 2016), Leibniz international proceedings in informatics (LIPIcs), vol 57, pp 49:1–49:15Google Scholar
- Jansen K, Rohwedder L (2017a) On the configuration-LP of the restricted assignment problem. In: Proceedings of the twenty-eighth annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 2670–2678Google Scholar
- Jansen K, Rohwedder L (2017b) A quasi-polynomial approximation for the restricted assignment problem. In: International conference on integer programming and combinatorial optimization, Springer, pp 305–316Google Scholar
- Jansen K, Land K, Maack M (2016) Estimating the makespan of the two-valued restricted assignment problem. In: LIPIcs-Leibniz international proceedings in informatics, Schloss Dagstuhl–Leibniz–Zentrum fuer Informatik, vol 53Google Scholar
- Kolliopoulos S, Moysoglou Y (2013) The 2-valued case of makespan minimization with assignment constraints. Inf Process Lett 113(1):39–43MathSciNetCrossRefzbMATHGoogle Scholar
- Lenstra J, Shmoys D, Tardos E (1990) Approximation algorithms for scheduling unrelated parallel machines. Math Program 46(1–3):259–271MathSciNetCrossRefzbMATHGoogle Scholar
- Lin Y, Li W (2004) Parallel machine scheduling of machine-dependent jobs with unit-length. Eur J Oper Res 156(1):261–266MathSciNetCrossRefzbMATHGoogle Scholar
- Page D, Solis-Oba R (2016) A 3/2-approximation algorithm for the graph balancing problem with two weights. Algorithms 9(2):38MathSciNetCrossRefGoogle Scholar
- Shchepin E, Vakhania N (2005) An optimal rounding gives a better approximation for scheduling unrelated machines. Oper Res Lett 33:127–133MathSciNetCrossRefzbMATHGoogle Scholar
- Shmoys D, Tardos É (1993) An approximation algorithm for the generalized assignment problem. Math Program 62(1–3):461–474MathSciNetCrossRefzbMATHGoogle Scholar
- Svensson O (2011) Santa Claus schedules jobs on unrelated machines. In: STOC’11 Proceedings of the 43rd ACM symposium on theory of computing, ACM, New York, pp 617–626Google Scholar
- Verschae J, Wiese A (2014) On the configuration-LP for scheduling on unrelated machines. J Sched 17(4):371–383MathSciNetCrossRefzbMATHGoogle Scholar
- Wang C, Sitters R (2016) On some special cases of the restricted assignment problem. Inf Proces Lett 116(11):723–728MathSciNetCrossRefzbMATHGoogle Scholar
- Williamson D, Shmoys D (2011) The design of approximation algorithms. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar