Abstract
This paper addresses the max–min \(k_i\)-partitioning problem that asks for an assignment of n jobs to m parallel machines so that the minimum machine completion time is maximized and the number of jobs on each machine does not exceed a machine-dependent cardinality limit \(k_i\) \((i=1,\ldots ,m)\). We propose different preprocessing as well as lifting procedures and derive several upper bound arguments. Furthermore, we introduce suited construction heuristics as well as an effective dynamic programming based improvement procedure. Results of a comprehensive computational study on a large set of randomly generated instances indicate that our algorithm quickly finds (near-)optimal solutions.
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Babel, L., Kellerer, H., Kotov, V.: The \(k\)-partitioning problem. Math. Methods Oper. Res. 47, 59–82 (1998)
Chen, S.P., He, Y., Yao, E.-Y.: Three-partitioning containing kernels: complexity and heuristic. Computing 57, 255–271 (1996)
Chen, S.P., He, Y., Lin, G.: 3-Partitioning problems for maximizing the minimum load. J. Comb. Optim. 6, 67–80 (2002)
Csirik, J., Kellerer, H., Woeginger, G.: The exact LPT-bound for maximizing the minimum completion time. Oper. Res. Lett. 11, 281–287 (1992)
Dell’Amico, M., Martello, S.: Optimal scheduling of tasks on identical parallel processors. ORSA J. Comput. 7, 191–200 (1995)
Dell’Amico, Martello, S.: Bounds for the cardinality constrained \(P, : C_{\rm max}\) problem. J. Sched. 4, 123–138 (2001)
Dell’Amico, M., Iori, M., Martello, S.: Heuristic algorithms and scatter search for the cardinality constrained \(P||C_{{\rm max}}\) problem. J. Heuristics 10, 169–204 (2004)
Dell’Amico, M., Iori, M., Martello, S., Monaci, M.: Lower bounds and heuristics for the \(k_i\)-partitioning problem. Eur. J. Oper. Res. 171, 725–742 (2006)
Deuermeyer, B.L., Friesen, D.K., Langston, M.A.: Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM J. Algebr. Discrete Methods 3, 190–196 (1982)
Friesen, D.K., Deuermeyer, B.L.: Analysis of greedy solutions for a replacement part sequencing problem. Math. Oper. Res. 6, 74–87 (1981)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)
Haouari, M., Jemmali, M.: Maximizing the minimum completion time on parallel machines. 4OR A Q. J. Oper. Res. 6, 375–392 (2008)
He, Y., Tan, Z., Zhu, J., Yao, E.: \(k\)-Partitioning problems for maximizing the minimum load. Comput. Math. Appl. 46, 1671–1681 (2003)
Kellerer, H., Kotov, V.: A 7/6-approximation algorithm for 3-partitioning and its application to multiprocessor scheduling. INFOR 37, 48–56 (1999)
Kellerer, H., Kotov, V.: A 3/2-approximation algorithm for \(k_i\)-partitioning. Oper. Res. Lett. 39, 359–362 (2011)
Kellerer, H., Woeginger, G.J.: A tight bound for 3-partitioning. Discrete Appl. Math. 45, 249–259 (1993)
Mertens, S.: A complete anytime algorithm for balanced number partitioning. Preprint arXiv:cs/9903011v1 (1999)
Michiels, W., Aarts, E., Korst, J., van Leeuwen, J., Spieksma, F.C.R.: Computer-assisted proof of performance ratios for the differencing method. Discrete Optim. 9, 1–16 (2012)
Tsai, L.-H.: Asymptotic analysis of an algorithm for balanced parallel processor scheduling. SIAM J. Comput. 21, 59–64 (1992)
Walter, R.: Comparing the minimum completion times of two longest-first scheduling-heuristics. CEJOR 21, 125–139 (2013)
Walter, R., Wirth, M., Lawrinenko, A.: Improved approaches to the exact solution of the machine covering problem. J. Sched. 20, 147–164 (2017)
Woeginger, G.J.: A polynomial-time approximation scheme for maximizing the minimum machine completion time. Oper. Res. Lett. 20, 149–154 (1997)
Woeginger, G.J.: A comment on scheduling two parallel machines with capacity constraints. Discrete Optim. 2, 269–272 (2005)
Zhang, C., Wang, G., Liu, X., Liu, J.: Approximating scheduling machines with capacity constraints. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) Frontiers in Algorithmics. Lecture Notes in Computer Science 5598, pp. 283–292. Springer, Berlin (2009)
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Lawrinenko, A., Schwerdfeger, S. & Walter, R. Reduction criteria, upper bounds, and a dynamic programming based heuristic for the max–min \(k_i\)-partitioning problem. J Heuristics 24, 173–203 (2018). https://doi.org/10.1007/s10732-017-9362-9
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DOI: https://doi.org/10.1007/s10732-017-9362-9