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Three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time

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Abstract

In this paper, we provide three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time. First, we propose an exact algorithm for in-trees, of which the complexity depends mainly on the graph height, i.e., the length of the longest chain of the precedence graph. We show that this work improves the algorithm in the literature both theoretically and experimentally. Second, we close the open problem for level-orders by showing how it is polynomially solvable. Third, we prove that preemptive scheduling in-trees is strongly NP-hard with arbitrary number of machines, of which the complexity was also open.

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Authors and Affiliations

  1. School of Economics and Management, Beihang University, Beijing, China

    Tianyu Wang

  2. IMT Atlantique, LS2N, La Chantrerie, 4 rue Alfred Kastler, 44307, Nantes, France

    Odile Bellenguez

Authors
  1. Tianyu Wang
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  2. Odile Bellenguez
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Correspondence to Tianyu Wang.

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Appendix

Appendix

The candidate sets of the example in Fig. 6. {1}, {2}, {3}, {4}, {5}, {8}, {9}, {10}, {11}, {12}, {13}, {14}, {15}, {1, 2}, {1, 3}, {1, 8}, {1, 9}, {1, 10}, {1, 12}, {1, 13}, {2, 3}, {2, 12}, {2, 13}, {3, 12}, {3, 13}, {4, 5}, {4, 11}, {5, 11}, {8, 11}, {8, 12}, {8, 9}, {8, 12}, {8, 11}, {8, 14}, {9, 10}, {9, 11}, {9, 12}, {9, 14}, {10, 11}, {10,12}, {10, 13}, {10, 14}, {11, 12}, {11, 13}, {12, 13}, {13, 14}.

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Wang, T., Bellenguez, O. Three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time. J Sched 24, 649–662 (2021). https://doi.org/10.1007/s10951-021-00702-w

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