Abstract
In this paper we study the single machine scheduling problem with release dates, deadlines and precedence relations where the objective is to minimize the makespan. This is a well-known strongly NP-hard scheduling problem [18]. We analyze the problem from the parameterized complexity point of view. We propose parameter \(q\) which is the maximum number of time windows \([r_j, d_j)\) that can strictly include a time window \([r_i, d_i)\) on both ends. We show that problems \(1|prec, r_j, d_j|C_{max}\) and \(1|prec, r_j|L_{max}\) are fixed-parameter tractable parameterized by \(q\). We use a dynamic programming approach and define a new dominance rule, which we call the weak earliest deadline rule. This rule narrows down the number of relevant scheduling prefixes enough to complete the search via a fixed-parameter tractable number of dynamic programming states.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Baart, R., de Weerdt, M., He, L.: Single-machine scheduling with release times, deadlines, setup times, and rejection. Eur. J. Oper. Res. 291(2), 629–639 (2021)
Baker, K.R.: The effects of input control in a simple scheduling model. J. Oper. Manag. 4(2), 99–112 (1984)
Bodlaender, H.L., Fellows, M.R.: W[2]-hardness of precedence constrained k-processor scheduling. Oper. Res. Lett. 18(2), 93–97 (1995)
Bodlaender, H.L., van der Wegen, M.: Parameterized complexity of scheduling chains of jobs with delays. In: Cao, Y., Pilipczuk, M. (eds.) 15th International Symposium on Parameterized and Exact Computation (IPEC), December 14-18 2020, Hong Kong, China (Virtual Conference), LIPIcs, vol. 180, pp. 1–15 (2020)
Bredereck, R., Bulteau, L., Komusiewicz, C., Talmon, N., van Bevern, R., Woeginger, G.J.: Precedence-constrained scheduling problems parameterized by partial order width. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) Discrete Optimization and Operations Research. Lecture Notes in Computer Science(), vol. 9869, pp. 105–120. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_9
Downey, R., Fellows, M.: Parameterized Complexity. Springer, Cham (1999)
Downey, R.G., Fellows, M.R., Regan, K.W.: Descriptive complexity and the W hierarchy. In: Proof Complexity and Feasible Arithmetics, pp. 119–134 (1996)
Erschler, J., Fontan, G., Merce, C.: Un nouveau concept de dominance pour l’ordonnancement de travaux sur une machine. RAIRO-Oper. Res. 19(1), 15–26 (1985)
Erschler, J., Fontan, G., Mercé, C., Roubellat, F.: A new dominance concept in scheduling n jobs on a single machine with ready times and due dates. Oper. Res. 31(1), 114–127 (1983). https://doi.org/10.1287/OPRE.31.1.114
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Cham (1998)
Gordon, V.S., Werner, F., Yanushkevich, O.: Single machine preemptive scheduling to minimize the weighted number of late jobs with deadlines and nested release/due date intervals. RAIRO-Oper. Res. 35(1), 71–83 (2001)
Grigoreva, N.: Single machine scheduling with precedence constrains, release and delivery times. In: Wilimowska, Z., Borzemski, L., Swiatek, J. (eds.) Information Systems Architecture and Technology: Proceedings of 40th Anniversary International Conference on Information Systems Architecture and Technology – ISAT 2019. Advances in Intelligent Systems and Computing, vol. 1052, pp. 188–198. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-30443-0_17
Hall, L.A., Shmoys, D.B.: Jackson’s rule for single-machine scheduling: Making a good heuristic better. Math. Oper. Res. 17(1), 22–35 (1992)
Hanen, C., Munier Kordon, A.: Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines. J. Sched., 1–15 (2023)
Hermelin, D., Karhi, S., Pinedo, M., Shabtay, D.: New algorithms for minimizing the weighted number of tardy jobs on a single machine. Ann. Oper. Res. 298(1), 271–287 (2021)
Kordon, A.M., Tang, N.: A fixed-parameter algorithm for scheduling unit dependent tasks with unit communication delays. In: Sousa, L., Roma, N., Tomas, P. (eds.) Euro-Par 2021: Parallel Processing. Lecture Notes in Computer Science(), vol. 12820, pp. 105–119. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85665-6_7
Lawler, E.L.: Optimal sequencing of a single machine subject to precedence constraints. Manage. Sci. 19(5), 544–546 (1973)
Lenstra, J., Rinnooy Kan, A., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977)
Mallem, M., Hanen, C., Munier-Kordon, A.: Parameterized complexity of a parallel machine scheduling problem. In: 17th International Symposium on Parameterized and Exact Computation (IPEC) (2022)
Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1), 533–562 (2015)
Proskurowski, A., Telle, J.A.: Classes of graphs with restricted interval models. Discrete Math. Theor. Comput. Sci. 3 (1999)
Sourd, F., Nuijten, W.: Scheduling with tails and deadlines. J. Sched. 4(2), 105–121 (2001)
Ware, E.B.: Job shop simulation on the IBM 704. In: Preprints of Papers Presented at the 14th National Meeting of the Association for Computing Machinery (1959)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Mallem, M., Hanen, C., Munier-Kordon, A. (2024). A New Structural Parameter on Single Machine Scheduling with Release Dates and Deadlines. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-031-60924-4_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-60923-7
Online ISBN: 978-3-031-60924-4
eBook Packages: Computer ScienceComputer Science (R0)