On the power of randomization for job shop scheduling with $k$-units length tasks

Mömke, Tobias

On the power of randomization for job shop scheduling with

k

-units length tasks

Mömke, Tobias

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009), p. 189-207 / Harvested from Numdam

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Résumé

In the job shop scheduling problem $k$ -units- $J_{m}$ , there are $m$ machines and each machine has an integer processing time of at most $k$ time units. Each job consists of a permutation of $m$ tasks corresponding to all machines and thus all jobs have an identical dilation $D$ . The contribution of this paper are the following results; (i) for $d = o (\sqrt{D})$ jobs and every fixed $k$ , the makespan of an optimal schedule is at most $D + o (D)$ , which extends the result of [3] for $k = 1$ ; (ii) a randomized on-line approximation algorithm for $k$ -units- $J_{m}$ is presented. This is the on-line algorithm with the best known competitive ratio against an oblivious adversary for $d = o (\sqrt{D})$ and $k > 1$ ; (iii) different processing times yield harder instances than identical processing times. There is no $5 / 3$ competitive deterministic on-line algorithm for $k$ -units- $J_{m}$ , whereas the competitive ratio of the randomized on-line algorithm of (ii) still tends to 1 for $d = o (\sqrt{D})$ .

Publié le : 2009-01-01

MR 2512254 | Zbl 1166.68041 | 1 citation dans Numdam

DOI : https://doi.org/10.1051/ita:2008024
Classification: 68W20, 68W25

@article{ITA_2009__43_2_189_0,
     author = {M\"omke, Tobias},
     title = {On the power of randomization for job shop scheduling with $k$-units length tasks},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {43},
     year = {2009},
     pages = {189-207},
     doi = {10.1051/ita:2008024},
     mrnumber = {2512254},
     zbl = {1166.68041},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2009__43_2_189_0}
}

Mömke, Tobias. On the power of randomization for job shop scheduling with $k$-units length tasks. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) pp. 189-207. doi : 10.1051/ita:2008024. http://gdmltest.u-ga.fr/item/ITA_2009__43_2_189_0/

Bibliographie

[1] P. Brucker, An efficient algorithm for the job shop problem with two jobs. Computing 40 (1988) 353-359. | MR 969653 | Zbl 0654.90036

[2] U. Feige and C. Scheideler, Improved bounds for acyclic job shop scheduling. Combinatorica 22 (2002) 361-399. | MR 1932059

[3] J. Hromkovič, K. Steinhöfel and P. Widmayer, Job shop scheduling with unit length tasks: bounds and algorithms. ICTCS '01: Proceedings of the 7th Italian Conference on Theoretical Computer Science. Lect. Notes Comput. Sci. 2202 (2001) 90-106. | MR 1915409 | Zbl 1042.90020

[4] J. Hromkovič, T. Mömke, K. Steinhöfel and P. Widmayer, Job shop scheduling with unit length tasks: bounds and algorithms. Algorithmic Operations Research 2 (2007) 1-14. | MR 2302153 | Zbl pre05179132

[5] F.A. Leighton, B.M. Maggs and S.B. Rao, Packet routing and job shop scheduling in $O (congestion + dilition)$ steps. Combinatorica 14 (1994) 167-186. | MR 1289071 | Zbl 0811.68062

[6] F.A. Leighton, B.M. Maggs and A.W. Richa, Fast algorithms for finding $O (congestion + dilation)$ packet routing schedules. Combinatorica 19 (1999) 375-401. | MR 1723254 | Zbl 0932.68005

[7] J.K. Lenstra and A.H.G. Rinnooy Kan, Computational complexity of discrete optimization problems. Annals of Discrete Mathematics 4 (1979) 121-140. | MR 558557 | Zbl 0411.68042

[8] D.P. Willamson, L.A. Hall, C.A.J. Hurkens, J.K. Lenstra, S.V. Sevast'Janov and D.B. Shmoys, Short shop schedules. Operations Research 45 (1997) 288-294. | MR 1644998 | Zbl 0890.90112