SchedulingLPs bear probabilities randomized approximations for minsum criteria
 Andreas S. Schulz,
 Martin Skutella
 … show all 2 hide
Abstract
In this paper, we provide a new class of randomized approximation algorithms for scheduling problems by directly interpreting solutions to socalled timeindexed LPs as probabilities. The most general model we consider is scheduling unrelated parallel machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. The crucial idea for these multiple machine problems is not to use standard list scheduling but rather to assign jobs randomly to machines (with probabilities taken from an optimal LP solution) and to perform list scheduling on each of them.
For the general model, we give a (2+ E)approximation algorithm. The best previously known approximation algorithm has a performance guarantee of 16/3 [HSW96]. Moreover, our algorithm also improves upon the best previously known approximation algorithms for the special case of identical parallel machine scheduling (performance guarantee (2.89 + E) in general [CPS+96] and 2.85 for the average completion time [CMNS97], respectively). A perhaps surprising implication for identical parallel machines is that jobs are randomly assigned to machines, in which each machine is equally likely. In addition, in this case the algorithm has running time O(nlogn) and performance guarantee 2. The same algorithm also is a 2approximation for the corresponding preemptive scheduling problem on identical parallel machines.
Finally, the results for identical parallel machine scheduling apply to both the offline and the online settings with no difference in performance guarantees. In the online setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards.
 B. Awerbuch, S. Kutten, and D. Peleg. Competitive distributed job scheduling. In Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, pages 571–581, 1992.
 J. L. Bruno, E. G. Coffman Jr., and R. Sethi. Scheduling independent tasks to reduce mean finishing time. Communications of the Association for Computing Machinery, 17:382–387, 1974.
 C. S. Chekuri, R. Johnson, R. Motwani, B. K. Natarajan, B. R. Rau, and M. Schlansker. Profiledriven instruction level parallel scheduling with applications to super blocks. December 1996. Proceedings of the 29th Annual International Symposium on Microarchitecture (MICRO29), Paris, France.
 S. Chakrabarti and S. Muthukrishnan. Resource scheduling for parallel database and scientific applications. June 1996. Proceedings of the 8th ACM Symposium on Parallel Algorithms and Architectures.
 C. S. Chekurï, R. Motwani, B. Natarajan, and C. Stein. Approximation techniques for average completion time scheduling. In Proceedings of the 8th ACMSIAM Symposium on Discrete Algorithms, pages 609–618, 1997.
 S. Chakrabarti, C. A. Phillips, A. S. Schulz, D. B. Shmoys, C. Stein, and J. Wein. Improved scheduling algorithms for minsum criteria. In F. Meyer auf der Heide and B. Monien, editors, Automata, Languages and Programming, number 1099 in Lecture Notes in Computer Science, pages 646–657. Springer, Berlin, 1996. Proceedings of the 23rd International Colloquium (ICALP'96).
 F A. Chudak and D. B. Shmoys. Approximation algorithms for precedenceconstrained scheduling problems on parallel machines that run at different speeds. In Proceedings of the 8th ACMSIAM Symposium on Discrete Algorithms, pages 581–590,1997.
 X. Deng, H. Liu, J. Long, and B. Xiao. Deterministic load balancing in computer networks. In Proceedings of the 2nd Annual IEEE Symposium on Parallel and Distributed Processing, pages 50–57, 1990.
 M. E. Dyer and L. A. Wolsey. Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Applied Mathematics, 26:255–270, 1990.
 P. Erdas and J. L. Selfridge. On a combinatorial game. Journal of Combinatorial Theory A, 14:298–301, 1973.
 R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5:287–326, 1979.
 M. X. Goemans. A supermodular relaxation for scheduling with release dates. In W. H. Cunningham, S. T. McCormick, and M. Queyranne, editors, Integer Programming and Combinatorial Optimization, number 1084 in Lecture Notes in Computer Science, pages 288–300. Springer, Berlin, 1996. Proceedings of the 5th International IPCO Conference.
 M. X. Goemans. Improved approximation algorithms for scheduling with release dates. In Proceedings of the 8th ACMSIAM Symposium on Discrete Algorithms, pages 591–598, 1997.
 M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella, and Y Wang, 1997. In preparation.
 L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Offline and online approximation algorithms. Preprint 516/1996, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1996. To appear in Mathematics of Operations Research.
 L. A. Hall, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Offline and online algorithms. In Proceedings of the 7th ACMSIAM Symposium on Discrete Algorithms, pages 142–151, 1996.
 T. Kawaguchi and S. Kyan. Worst case bound of an LRF schedule for the mean weighted flowtime problem. SIAM Journal on Computing, 15:1119–1129, 1986.
 J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343–362, 1977.
 R. Motwani, J. Naor, and P Raghavan. Randomized approximation algorithms in combinatorial optimization. In D. S. Hochbaum, editor, Approximation Algorithms for NPHard Problems, chapter 11. Thomson, 1996.
 R. Motwani and P Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
 R. H. Möhring, M. W. Schäffter, and A. S. Schulz. Scheduling jobs with communication delays: Using infeasible solutions for approximation. In J. Diaz and M. Serna, editors, AlgorithmsESA'96, volume 1136 of Lecture Notes in Com puter Science, pages 76–90. Springer, Berlin, 1996. Proceedings of the 4th Annual European Symposium on Algorithms.
 C. Phillips, C. Stein, and J. Wein. Task scheduling in networks. In Algorithm TheorySWAT'94, number 824 in Lecture Notes in Computer Science, pages 290–301. Springer, Berlin, 1994. Proceedings of the 4th Scandinavian Workshop on Algorithm Theory.
 C. Phillips, C. Stein, and J. Wein. Scheduling jobs that arrive over time. In Proceedings of the Fourth Workshop on Algorithms and Data Structures, number 955 in Lecture Notes in Computer Science, pages 86–97. Springer, Berlin, 1995.
 P. Raghavan and C. D. Thompson. Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica, 7:365–374, 1987.
 A. S. Schulz. Scheduling to minimize total weighted completion time: Performance guarantees of LPbased heuristics and lower bounds. In W. H. Cunningham, S. T. McCormick, and M. Queyranne, editors, Integer Programming and Combinatorial Optimization, number 1084 in Lecture Notes in Computer Science, pages 301–315. Springer, Berlin, 1996. Proceedings of the 5th International IPCO Conference.
 J. Spencer. Ten Lectures on the Probabilistic Method. Number 52 in CBMSNSF Reg. Conf. Ser. Appl. Math. SIAM, 1987.
 A. S. Schulz and M. Skutella. Randombased scheduling: New approximations and LP lower bounds. Preprint 549/1997, Department of Mathematics, Technical University of Berlin, Berlin, Germany, February 1997. To appear in Springer Lecture Notes in Computer Science, Proceedings of the 1st International Symposium on Randomization and Approximation Techniques in Computer Science (Random' 97).
 D. B. Shmoys and É. Tardos. An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62:461–474, 1993.
 Title
 SchedulingLPs bear probabilities randomized approximations for minsum criteria
 Book Title
 Algorithms — ESA '97
 Book Subtitle
 5th Annual European Symposium Graz, Austria, September 15–17, 1997 Proceedings
 Pages
 pp 416429
 Copyright
 1997
 DOI
 10.1007/3540633979_32
 Print ISBN
 9783540633976
 Online ISBN
 9783540695363
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1284
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 eBook Packages
 Editors
 Authors

 Andreas S. Schulz ^{(1)}
 Martin Skutella ^{(1)}
 Author Affiliations

 1. Fachbereich Mathematik, Technische Universität Berlin, MA 61, Straße des 17. Juni 136, 10623, Berlin, Germany
Continue reading...
To view the rest of this content please follow the download PDF link above.