TY - JOUR
T1 - Scheduling lower bounds via AND subset sum
AU - Abboud, Amir
AU - Bringmann, Karl
AU - Hermelin, Danny
AU - Shabtay, Dvir
N1 - Funding Information:
This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 850979).This research was supported by The Israel Science Foundation (Grant no. 1070/20).
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/8/1
Y1 - 2022/8/1
N2 - Given N instances (X1,t1),…,(XN,tN) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers Xi has a subset that sums up to the target integer ti. We prove that this problem cannot be solved in time O˜((N⋅tmax)1−ε), for tmax=maxiti and any ε>0, assuming the ∀∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude O˜(n+pmax⋅n1−ε)-time algorithms for several scheduling problems on n jobs with maximum processing time pmax, assuming ∀∃-SETH. These include classical problems such as 1||∑wjUj, the problem of minimizing the total weight of tardy jobs on a single machine, and P2||∑Uj, the problem of minimizing the number of tardy jobs on two identical parallel machines.
AB - Given N instances (X1,t1),…,(XN,tN) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers Xi has a subset that sums up to the target integer ti. We prove that this problem cannot be solved in time O˜((N⋅tmax)1−ε), for tmax=maxiti and any ε>0, assuming the ∀∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude O˜(n+pmax⋅n1−ε)-time algorithms for several scheduling problems on n jobs with maximum processing time pmax, assuming ∀∃-SETH. These include classical problems such as 1||∑wjUj, the problem of minimizing the total weight of tardy jobs on a single machine, and P2||∑Uj, the problem of minimizing the number of tardy jobs on two identical parallel machines.
KW - Lower bounds
KW - Parallel machine problems
KW - SETH
KW - Scheduling
KW - Single machine problems
KW - Subset sum
UR - http://www.scopus.com/inward/record.url?scp=85124699827&partnerID=8YFLogxK
U2 - 10.1016/j.jcss.2022.01.005
DO - 10.1016/j.jcss.2022.01.005
M3 - Article
AN - SCOPUS:85124699827
VL - 127
SP - 29
EP - 40
JO - Journal of Computer and System Sciences
JF - Journal of Computer and System Sciences
SN - 0022-0000
ER -