TY - GEN
T1 - A Simpler QPTAS for Scheduling Jobs with Precedence Constraints
AU - Das, Syamantak
AU - Wiese, Andreas
N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.
PY - 2022/9/1
Y1 - 2022/9/1
N2 - We study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a (2 - 1 m)-approximation. Interestingly, it is open whether the problem is NP-hard if m = 3 which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that m is a constant. In a breakthrough paper, Levey and Rothvoss presented a (1 + ϵ)-approximation with a running time of n(log n)O((m2/∈2) log log n) [STOC 2016, SICOMP 2019] and this running time was improved to quasipolynomial by Garg [ICALP 2018] and to even nOm,∈(log3 log n) by Li [SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules. In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly.
AB - We study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a (2 - 1 m)-approximation. Interestingly, it is open whether the problem is NP-hard if m = 3 which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that m is a constant. In a breakthrough paper, Levey and Rothvoss presented a (1 + ϵ)-approximation with a running time of n(log n)O((m2/∈2) log log n) [STOC 2016, SICOMP 2019] and this running time was improved to quasipolynomial by Garg [ICALP 2018] and to even nOm,∈(log3 log n) by Li [SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules. In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly.
KW - QPTAS
KW - makespan minimization
KW - precedence constraints
UR - http://www.scopus.com/inward/record.url?scp=85137563596&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ESA.2022.40
DO - 10.4230/LIPIcs.ESA.2022.40
M3 - Conference contribution
AN - SCOPUS:85137563596
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 30th Annual European Symposium on Algorithms, ESA 2022
A2 - Chechik, Shiri
A2 - Navarro, Gonzalo
A2 - Rotenberg, Eva
A2 - Herman, Grzegorz
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 30th Annual European Symposium on Algorithms, ESA 2022
Y2 - 5 September 2022 through 9 September 2022
ER -