TY - GEN
T1 - A (2 + ?)-approximation algorithm for preemptive weighted flow time on a single machine
AU - Rohwedder, Lars
AU - Wiese, Andreas
N1 - Publisher Copyright:
© 2021 ACM.
PY - 2021/6/15
Y1 - 2021/6/15
N2 - Weighted flow time is a fundamental and very well-studied objective function in scheduling. In this paper, we study the setting of a single machine with preemptions. The input consists of a set of jobs, characterized by their processing times, release times, and weights and we want to compute a (possibly preemptive) schedule for them. The objective is to minimize the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its release date and its completion time. It had been a long-standing open problem to find a polynomial time O(1)-approximation algorithm for this setting. In a recent break-through result, Batra, Garg, and Kumar (FOCS 2018) found such an algorithm if the input data are polynomially bounded integers, and Feige, Kulkarni, and Li (SODA 2019) presented a black-box reduction to this setting. The resulting approximation ratio is a (not explicitly stated) constant which is at least 10000. In this paper we improve this ratio to 2+?. The algorithm by Batra, Garg, and Kumar (FOCS 2018) reduces the problem to Demand MultiCut on trees and solves the resulting instances via LP-rounding and a dynamic program. Instead, we first reduce the problem to a (different) geometric problem while losing only a factor 1+?, and then solve its resulting instances up to a factor of 2+? by a dynamic program. In particular, our reduction ensures certain structural properties, thanks to which we do not need LP-rounding methods. We believe that our result makes substantial progress towards finding a PTAS for weighted flow time on a single machine.
AB - Weighted flow time is a fundamental and very well-studied objective function in scheduling. In this paper, we study the setting of a single machine with preemptions. The input consists of a set of jobs, characterized by their processing times, release times, and weights and we want to compute a (possibly preemptive) schedule for them. The objective is to minimize the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its release date and its completion time. It had been a long-standing open problem to find a polynomial time O(1)-approximation algorithm for this setting. In a recent break-through result, Batra, Garg, and Kumar (FOCS 2018) found such an algorithm if the input data are polynomially bounded integers, and Feige, Kulkarni, and Li (SODA 2019) presented a black-box reduction to this setting. The resulting approximation ratio is a (not explicitly stated) constant which is at least 10000. In this paper we improve this ratio to 2+?. The algorithm by Batra, Garg, and Kumar (FOCS 2018) reduces the problem to Demand MultiCut on trees and solves the resulting instances via LP-rounding and a dynamic program. Instead, we first reduce the problem to a (different) geometric problem while losing only a factor 1+?, and then solve its resulting instances up to a factor of 2+? by a dynamic program. In particular, our reduction ensures certain structural properties, thanks to which we do not need LP-rounding methods. We believe that our result makes substantial progress towards finding a PTAS for weighted flow time on a single machine.
KW - approximation algorithm
KW - dynamic programming
KW - scheduling
UR - https://www.scopus.com/pages/publications/85108161996
U2 - 10.1145/3406325.3451075
DO - 10.1145/3406325.3451075
M3 - Conference contribution
AN - SCOPUS:85108161996
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1042
EP - 1055
BT - STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
A2 - Khuller, Samir
A2 - Williams, Virginia Vassilevska
PB - Association for Computing Machinery
T2 - 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Y2 - 21 June 2021 through 25 June 2021
ER -