TY - JOUR

T1 - Faster minimization of tardy processing time on a single machine

AU - Bringmann, Karl

AU - Fischer, Nick

AU - Hermelin, Dan

AU - Shabtay, Dvir

AU - Wellnitz, Philip

N1 - Publisher Copyright: © Karl Bringmann, Nick Fischer, Danny Hermelin, Dvir Shabtay, and Philip Wellnitz; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - This paper is concerned with the 1||P 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 a very 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 fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs 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 1||P pjUj, each improving on Lawler and Moore's algorithm in a different scenario: Our first algorithm runs in Õ(P7/4) time1, and outperforms Lawler and Moore's algorithm in instances where n = ω~(P3/4). Our second algorithm runs in Õ(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 Õ(P) algorithm for the single due date special case of 1||P 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, while for the first algorithm we define a new “skewed” version of (max, min)convolution which is interesting in its own right.

AB - This paper is concerned with the 1||P 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 a very 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 fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs 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 1||P pjUj, each improving on Lawler and Moore's algorithm in a different scenario: Our first algorithm runs in Õ(P7/4) time1, and outperforms Lawler and Moore's algorithm in instances where n = ω~(P3/4). Our second algorithm runs in Õ(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 Õ(P) algorithm for the single due date special case of 1||P 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, while for the first algorithm we define a new “skewed” version of (max, min)convolution which is interesting in its own right.

KW - Convolutions

KW - Sumsets

KW - Weighted number of tardy jobs

UR - http://www.scopus.com/inward/record.url?scp=85089353459&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ICALP.2020.19

DO - https://doi.org/10.4230/LIPIcs.ICALP.2020.19

M3 - مقالة من مؤنمر

SN - 1868-8969

VL - 168

JO - Leibniz International Proceedings in Informatics, LIPIcs

JF - Leibniz International Proceedings in Informatics, LIPIcs

T2 - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

Y2 - 8 July 2020 through 11 July 2020

ER -