# Algorithms and Approximation Schemes for Minimum Lateness/Tardiness Scheduling with Rejection

## Abstract

We consider the problem of minimum lateness/tardiness scheduling with rejection for which the objective function is the sum of the maximum lateness/tardiness of the scheduled jobs and the total rejection penalty (sum of rejection costs) of the rejected jobs. If rejection is not considered, the problems are solvable in polynomial time using the *Earliest Due Date (EDD)* rule. We show that adding the option of rejection makes the problems \(\mathcal{NP}\)-complete. We give pseudo-polynomial time algorithms, based on dynamic programming, for these problems. We also develop a fully polynomial time approximation scheme (FPTAS) for minimum tardiness scheduling with rejection using a geometric rounding technique on the total rejection penalty.

Observe that the usual notion of an approximation algorithm (guaranteed factor bound relative to optimal objective function value) is inappropriate when the optimal objective function value could be negative, as is the case with minimum lateness scheduling with rejection. An alternative notion of approximation, called *ε*-*optimization approximation* [7], is suitable for designing approximation algorithms for such problems. We give a polynomial time *ε*-optimization approximation scheme (PTEOS) for minimum lateness scheduling with rejection and a fully polynomial time *ε*-optimization approximation scheme (FPTEOS) for a modified problem where the total rejection penalty is the *product* (and not the sum) of the rejection costs of the rejected jobs.

## Keywords

Dynamic Program Polynomial Time Optimal Schedule Maximum Lateness Partition Problem## Preview

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## References

- 1.Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics 5, 287–326 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
- 2.Lawler, E.L., Lenstra, J.K., Rinooy Kan, A.H.G., Shmoys, D.B.: Sequencing and Scheduling: Algorithms and Complexity. In: Handbooks in Operations Research and Management Science. Logistics of Production and Inventory, vol. 4, pp. 445–522. North-Holland, Amsterdam (1993)Google Scholar
- 3.Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., Stougie, L.: Multiprocessor scheduling with rejection. In: 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 95–103 (1996)Google Scholar
- 4.Hoogeveen, H., Skutella, M., Woeginger, G.J.: Preemptive scheduling with rejection. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 268–277. Springer, Heidelberg (2000)CrossRefGoogle Scholar
- 5.Seiden, S.: Preemptive multiprocessor scheduling with rejection. Theoretical Computer Science 262(1), 437–458 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Engels, D.W., Karger, D.R., Kolliopoulos, S.G., Sengupta, S., Uma, R.N., Wein, J.: Techniques for Scheduling with Rejection. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 490–501. Springer, Heidelberg (1998)CrossRefGoogle Scholar
- 7.Orlin, J.B., Schulz, A.S., Sengupta, S.: ∈-Optimization Schemes and L-Bit Precision: Alternative Perspectives in Combinatorial Optimization. 32nd Annual ACM Symposium on Theory of Computing (STOC) (2000) Google Scholar
- 8.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar