Abstract
In the moldable job scheduling problem one has to assign a set of n jobs to m machines, in order to minimize the time it takes to process all jobs. Each job is moldable, so it can be assigned not only to one but any number of the identical machines. We assume that the work of each job is monotone and that jobs can be placed non-contiguously. In this work we present a \((\frac{3}{2} + \epsilon )\)-approximation algorithm with a worst-case runtime of \({O(n \log ^2(\frac{1}{\epsilon }+ \frac{\log (\epsilon m)}{\epsilon }) + \frac{n}{\epsilon } \log (\frac{1}{\epsilon }) {\log (\epsilon m)})}\) when \(m\le 16n\). This is an improvement over the best known algorithm of the same quality by a factor of \(\frac{1}{\epsilon }\) and several logarithmic dependencies. We complement this result with an improved FPTAS with running time \(O(n \log ^2(\frac{1}{\epsilon }+ \frac{\log (\epsilon m)}{\epsilon }))\) for instances with many machines \(m> 8\frac{n}{\epsilon }\). This yields a \(\frac{3}{2}\)-approximation with runtime \(O(n \log ^2(\log m))\) when \(m>16n\).
We achieve these results through one new core observation: In an approximation setting one does not need to consider all m possible allotments for each job. We will show that we can reduce the number of relevant allotments for each job from m to \(O(\frac{1}{\epsilon }+ \frac{\log (\epsilon m)}{\epsilon })\). Using this observation immediately yields the improved FPTAS. For the other result we use a reduction to the knapsack problem first introduced by Mounié, Rapine and Trystram. We use the reduced number of machines to give a new elaborate rounding scheme and define a modified version of this knapsack instance. This in turn allows for the application of a convolution based algorithm by Axiotis and Tzamos. We further back our theoretical results through a practical implementation and compare our algorithm to the previously known best result. These experiments show that our algorithm is faster and generates better solutions.
Supported by DFG-Project JA 612 /25-1.
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Notes
- 1.
We mainly consider 0-1 Knapsack, though some items may appear multiple times.
References
Axiotis, K., Tzamos, C.: Capacitated dynamic programming: faster knapsack and graph algorithms. In: 46th International Colloquium on Automata, Languages, and Programming (ICALP), Dagstuhl, Germany, vol. 132, pp. 19:1–19:13 (2019)
Bateni, M., Hajiaghayi, M., Seddighin, S., Stein, C.: Fast algorithms for knapsack via convolution and prediction. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, New York, NY, USA, pp. 1269–1282 (2018)
Belkhale, K.P., Banerjee, P.: An approximate algorithm for the partitionable independent task scheduling problem. In: International Conference on Parallel Processing (ICPP), pp. 72–75 (1990)
Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)
Cygan, M., Mucha, M., Wegrzycki, K., Wlodarczyk, M.: On problems equivalent to (min, +)-convolution. In: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). LIPIcs, vol. 80, pp. 22:1–22:15 (2017)
Drozdowski, M.: On the complexity of multiprocessor task scheduling. Bull. Pol. Acad. Sci.-Tech. Sci. 43, 381–392 (1995)
Du, J., Leung, J.Y.T.: Complexity of scheduling parallel task systems. SIAM J. Discret. Math. 2(4), 473–487 (1989)
Eisenbrand, F., Weismantel, R.: Proximity results and faster algorithms for integer programming using the Steinitz lemma. ACM Trans. Algorithms 16(1), 1–14 (2019)
Garey, M.R., Graham, R.L.: Bounds for multiprocessor scheduling with resource constraints. SIAM J. Comput. 4, 187–200 (1975)
Grage, K., Jansen, K., Ohnesorge, F.: Improved algorithms for monotone moldable job scheduling using compression and convolution (2023). https://arxiv.org/abs/2303.01414
Jansen, K., Land, F.: Scheduling monotone moldable jobs in linear time. In: 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 172–181. IEEE Computer Society, Los Alamitos (2018)
Jansen, K., Land, F., Land, K.: Bounding the running time of algorithms for scheduling and packing problems. Bericht des Instituts für Informatik, vol. 1302 (2013)
Ludwig, W., Tiwari, P.: Scheduling malleable and nonmalleable parallel tasks. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1994, pp. 167–176. Society for Industrial and Applied Mathematics, USA (1994)
Mounié, G., Rapine, C., Trystram, D.: A 3/2-dual approximation for scheduling independant monotonic malleable tasks. SIAM J. Comput. 37, 401–412 (2007)
Polak, A., Rohwedder, L., Węgrzycki, K.: Knapsack and subset sum with small items. In: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), vol. 198, pp. 106:1–106:19 (2021)
Turek, J., Wolf, J.L., Yu, P.S.: Approximate algorithms scheduling parallelizable tasks. In: Proceedings of the Fourth Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA 1992, pp. 323–332. Association for Computing Machinery, New York (1992)
Wu, F., Zhang, X., Chen, B.: An improved approximation algorithm for scheduling monotonic moldable tasks. Eur. J. Oper. Res. 306(2), 567–578 (2023)
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Grage, K., Jansen, K., Ohnesorge, F. (2023). Improved Algorithms for Monotone Moldable Job Scheduling Using Compression and Convolution. In: Cano, J., Dikaiakos, M.D., Papadopoulos, G.A., Pericàs, M., Sakellariou, R. (eds) Euro-Par 2023: Parallel Processing. Euro-Par 2023. Lecture Notes in Computer Science, vol 14100. Springer, Cham. https://doi.org/10.1007/978-3-031-39698-4_34
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