Abstract
We consider an online job scheduling problem on a single machine with precedence constraints under uncertainty. In this problem, for each trial \(t=1,\dots ,T\), the player chooses a total order (permutation) of n fixed jobs satisfying some prefixed precedence constraints. Then, the adversary determines the processing time for each job, 9 and the player incurs as loss the sum of the processing time and the waiting time. The goal of the player is to perform as well as the best fixed total order of jobs in hindsight. We formulate the problem as an online linear optimization problem over the permutahedron (the convex hull of permutation vectors) with specific linear constraints, in which the underlying decision space is written with exponentially many linear constraints. We propose a polynomial time online linear optimization algorithm; it predicts almost as well as the state-of-the-art offline approximation algorithms do in hindsight.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ailon, N.: Improved bounds for online learning over the permutahedron and other ranking polytopes. In: Proceedings of 17th International Conference on Artificial Intelligence and Statistics (AISTAT 2014), pp. 29–37 (2014)
Ambühl, C., Mastrolilli, M., Mutsanas, N., Svensson, O.: On the Approximability of Single-Machine Scheduling with Precedence Constraints Christoph Ambühl. Mathematics of Operations Research 36(4), 653–669 (2011)
Ambühl, C., Mastrolilli, M., Swensson, O.: Inapproximability results for sparsest cut, optimal linear arrangement, and precedence constrained scheduling. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 329–337 (2007)
Boykov, Y., Kolmogorov, V.: An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Computer Vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(9), 1124–1137 (2004)
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press (2006)
Chekuri, C., Motwani, R.: Precedence constrained scheduling to minimize sum of weighted completion times on a single machine. Discrete Applied Mathematics 98(1–2), 29–38 (1999)
Chudak, F.A., Hochbaum, D.S.: A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Operations Research Letters 25, 199–204 (1999)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Clifford, S.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)
Correa, J.R., Schulz, A.S.: Single-machine scheduling with precedence constraints. Mathematics of Operations Research 30(4), 1005–1021 (2005)
Even-Dar, E., Kleinberg, R., Mannor, S., Mansour, Y.: Online learning for global cost functions. In: Proceedings of the 22nd Conference on Learning Theory (COLT 2009) (2009)
Freund, Y., Schapire, R.E.: Large Margin Classification Using the Perceptron Algorithm. Machine Learning 37(3), 277–299 (1999)
Fujishige, S.: Submodular functions and optimization, 2nd edn. Elsevier Science (2005)
Fujita, T., Hatano, K., Takimoto, E.: Combinatorial online prediction via metarounding. In: Jain, S., Munos, R., Stephan, F., Zeugmann, T. (eds.) ALT 2013. LNCS, vol. 8139, pp. 68–82. Springer, Heidelberg (2013)
Hall, L.A., Schulz, A.S., Shmoys, D.B., Wein, J.: Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms. Mathematics of Operations Research 22(3), 513–544 (1997)
Helmbold, D.P., Warmuth, M.K.: Learning Permutations with Exponential Weights. Journal of Machine Learning Research 10, 1705–1736 (2009)
Kakade, S., Kalai, A.T., Ligett, L.: Playing games with approximation algorithms. SIAM Journal on Computing 39(3), 1018–1106 (2009)
Kalai, A., Vempala, S.: Efficient algorithms for online decision problems. Journal of Computer and System Sciences 71(3), 291–307 (2005)
Lawler, E.L.: On Sequencing jobs to minimize weighted completion time subject to precedence constraints. Annals of Discrete Mathematics 2(2), 75–90 (1978)
Lenstra, J.K., Kan, A.H.G.R.: Complexity of scheduling under precedence constraints. Operations Research 26, 22–35 (1978)
Lugosi, G., Papaspiliopoulos, O., Stoltz, G.: Online multi-task learning with hard constraints. In: Proceedings of the 22nd Conference on Learning Theory (COLT 2009) (2009)
Luss, R., Rosset, S., Shahar, M.: Efficient regularized isotonic regression with application to gene-gene interaction search. Annals of Applied Statistics 6(1) (2012)
Margot, F., Queyranne, M., Wang, Y.: Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling ProblemDecompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem. Operations Research 51(6), 981–992 (2003)
Maxwell, W., Muckstadt, J.: Establishing consistent and realistic reorder intervals in production-distribution systems. Operations Research 33, 1316–1341 (1985)
Mohring, H.R., Schulz, A.S., Uetz, M.: Approximation in Stochastic Scheduling: The Power of LP-based Priority Policies. Journal of the ACM 46(6), 924–942 (1999)
Schulz, A.S.: Scheduling to minimize total weighted completion time: performance guarantees of LP-based heuristics and lower bounds. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 301–315. Springer, Heidelberg (1996)
Skutella, M., Uetz, M.: Stochastic Machine Scheduling with Precedence Constraints. SIAM Journal on Computing 34(4), 788–802 (2005)
Spouge, J., Wan, H., Wilbur, W.: Least squares isotonic regression in two dimensions. J. Optimization Theory and Apps. 117, 585–605 (2003)
Suehiro, D., Hatano, K., Kijima, S., Takimoto, E., Nagano, K.: Online prediction under submodular constraints. In: Bshouty, N.H., Stoltz, G., Vayatis, N., Zeugmann, T. (eds.) ALT 2012. LNCS, vol. 7568, pp. 260–274. Springer, Heidelberg (2012)
Woeginger, G.J.: On the approximability of average completion time scheduling under precedence constraints. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 887–897. Springer, Heidelberg (2001)
Woeginger, G.J., Schuurman, P.: Polynomial time approximation algorithms for machine scheduling: Ten open problems. Journal of Scheduling 2, 203–213 (1999)
Yasutake, S., Hatano, K., Kijima, S., Takimoto, E., Takeda, M.: Online linear optimization over permutations. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 534–543. Springer, Heidelberg (2011)
Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag (1995)
Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Proceedings of the Twentieth International Conference on Machine Learning (ICML 2003), pp. 928–936 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Fujita, T., Hatano, K., Kijima, S., Takimoto, E. (2015). Online Linear Optimization for Job Scheduling Under Precedence Constraints. In: Chaudhuri, K., GENTILE, C., Zilles, S. (eds) Algorithmic Learning Theory. ALT 2015. Lecture Notes in Computer Science(), vol 9355. Springer, Cham. https://doi.org/10.1007/978-3-319-24486-0_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-24486-0_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24485-3
Online ISBN: 978-3-319-24486-0
eBook Packages: Computer ScienceComputer Science (R0)