Faster Minimization of Tardy Processing Time on a Single Machine

This paper is concerned with the 1 | | ∑ p j U j problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem...

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Vydáno v:	Algorithmica Ročník 84; číslo 5; s. 1341 - 1356
Hlavní autoři:	Bringmann, Karl, Fischer, Nick, Hermelin, Danny, Shabtay, Dvir, Wellnitz, Philip
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.05.2022 Springer Nature B.V
Témata:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Integers Knapsack problem Mathematics of Computing Multiplication Polynomials Run time (computers) Theory of Computation Tardy processing time min)-Convolution max Single machine scheduling Pseudo-polynomial time algorithm Fast polynomial multiplication
ISSN:	0178-4617, 1432-0541
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Abstract	This paper is concerned with the 1 \| \| ∑ p j U j problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version in O ( P · n ) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for problem, each improving on Lawler and Moore’s algorithm in a different scenario. Our first algorithm runs in O ~ ( P 7 / 4 ) time, and outperforms Lawler and Moore’s algorithm in instances where n = ω ~ ( P 3 / 4 ) . Our second algorithm runs in O ~ ( min { P · D # , P + D } ) time, where D # is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when n = ω ~ ( D # ) or n = ω ~ ( D / P ) . Further, it extends the known O ~ ( P ) algorithm for the single due date special case of 1 \| \| ∑ p j U j in a natural way. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new “skewed” version of ( max , min ) -Convolution which is interesting in its own right.
AbstractList	This paper is concerned with the 1\|\|∑pjUj problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version in O(P·n) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for problem, each improving on Lawler and Moore’s algorithm in a different scenario.Our first algorithm runs in O~(P7/4) time, and outperforms Lawler and Moore’s algorithm in instances where n=ω~(P3/4).Our second algorithm runs in O~(min{P·D#,P+D}) time, where D# is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when n=ω~(D#) or n=ω~(D/P). Further, it extends the known O~(P) algorithm for the single due date special case of 1\|\|∑pjUj in a natural way. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new “skewed” version of (max,min)-Convolution which is interesting in its own right. This paper is concerned with the 1 \| \| ∑ p j U j problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version in O ( P · n ) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for problem, each improving on Lawler and Moore’s algorithm in a different scenario. Our first algorithm runs in O ~ ( P 7 / 4 ) time, and outperforms Lawler and Moore’s algorithm in instances where n = ω ~ ( P 3 / 4 ) . Our second algorithm runs in O ~ ( min { P · D # , P + D } ) time, where D # is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when n = ω ~ ( D # ) or n = ω ~ ( D / P ) . Further, it extends the known O ~ ( P ) algorithm for the single due date special case of 1 \| \| ∑ p j U j in a natural way. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new “skewed” version of ( max , min ) -Convolution which is interesting in its own right.
Author	Hermelin, Danny Fischer, Nick Bringmann, Karl Wellnitz, Philip Shabtay, Dvir
Author_xml	– sequence: 1 givenname: Karl surname: Bringmann fullname: Bringmann, Karl organization: Saarland University, Saarland Informatics Campus (SIC), Max Planck Institute for Informatics, Saarland Informatics Campus (SIC) – sequence: 2 givenname: Nick surname: Fischer fullname: Fischer, Nick organization: Saarland University, Saarland Informatics Campus (SIC), Max Planck Institute for Informatics, Saarland Informatics Campus (SIC) – sequence: 3 givenname: Danny surname: Hermelin fullname: Hermelin, Danny organization: Department of Industrial Engineering and Management, Ben-Gurion University of the Negev – sequence: 4 givenname: Dvir surname: Shabtay fullname: Shabtay, Dvir email: dvirs@bgu.ac.il organization: Department of Industrial Engineering and Management, Ben-Gurion University of the Negev – sequence: 5 givenname: Philip surname: Wellnitz fullname: Wellnitz, Philip organization: Max Planck Institute for Informatics, Saarland Informatics Campus (SIC)
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References	Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer, Berlin (1972) Bringmann, K.: A near-linear pseudopolynomial time algorithm for subset sum. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1073–1084 (2017) Abboud, A., Bringmann, K., Hermelin, D., Shabtay, D.: SETH-based lower bounds for subset sum and bicriteria path. In: Proceedings of of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 41–57 (2019) Künnemann, M., Paturi, R., Schneider, S.: On the fine-grained complexity of one-dimensional dynamic programming. In: Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 21:1–21:15 (2017) PanVictor YSimple multivariate polynomial multiplicationJ. Symb. Comput.1994183183186131813310.1006/jsco.1994.1042 CyganMarekMuchaMarcinWegrzyckiKarolWlodarczykMichalOn problems equivalent to (min, +)-convolutionACM Trans. Algorithms201915114:114:2510.1145/3293465 CormenThomas HLeisersonCharles ERivestRonald LSteinCliffordIntroduction to Algorithms20093CambridgeThe MIT Press1187.68679 GrahamRonald LBounds on multiprocessing timing anomaliesSIAM J. Appl. Math.196917241642924921410.1137/0117039 Kosaraju, S.R.: Efficient tree pattern matching. In: Proceedings of the 30th annual symposium on Foundations Of Computer Science (FOCS), pp. 178–183 (1989) Koiliaris, K., Xu, C.: A faster pseudopolynomial time algorithm for subset sum. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1062–1072 (2017) LawlerEugene LMooreJames MA functional equation and its application to resource allocation and sequencing problemsManage. Sci.1969161778410.1287/mnsc.16.1.77 928_CR6 928_CR1 928_CR2 Ronald L Graham (928_CR5) 1969; 17 Thomas H Cormen (928_CR3) 2009 Victor Y Pan (928_CR11) 1994; 18 Marek Cygan (928_CR4) 2019; 15 Eugene L Lawler (928_CR10) 1969; 16 928_CR9 928_CR7 928_CR8
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Integers Knapsack problem Mathematics of Computing Multiplication Polynomials Run time (computers) Theory of Computation
Title	Faster Minimization of Tardy Processing Time on a Single Machine
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