Scheduling lower bounds via AND subset sum

Amir Abboud; Karl Bringmann; Danny Hermelin; Dvir Shabtay

doi:10.1016/j.jcss.2022.01.005

Scheduling lower bounds via AND subset sum

Amir Abboud, Karl Bringmann, Danny Hermelin, Dvir Shabtay

Department of Industrial Engineering and Management

Research output: Contribution to journal › Article › peer-review

Abstract

Given N instances (X₁,t₁),…,(X_N,t_N) 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 X_i has a subset that sums up to the target integer t_i. We prove that this problem cannot be solved in time O˜((N⋅t_max)^1−ε), for t_max=max_i⁡t_i and any ε>0, assuming the ∀∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude O˜(n+p_max⋅n^1−ε)-time algorithms for several scheduling problems on n jobs with maximum processing time p_max, assuming ∀∃-SETH. These include classical problems such as 1||∑w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P₂||∑U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.

Original language	English
Pages (from-to)	29-40
Number of pages	12
Journal	Journal of Computer and System Sciences
Volume	127
DOIs	https://doi.org/10.1016/j.jcss.2022.01.005
State	Published - 1 Aug 2022

Keywords

Lower bounds
Parallel machine problems
SETH
Scheduling
Single machine problems
Subset sum

ASJC Scopus subject areas

Theoretical Computer Science
Computer Networks and Communications
Computational Theory and Mathematics
Applied Mathematics

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10.1016/j.jcss.2022.01.005

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title = "Scheduling lower bounds via AND subset sum",

abstract = "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=maxi⁡ti 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.",

keywords = "Lower bounds, Parallel machine problems, SETH, Scheduling, Single machine problems, Subset sum",

author = "Amir Abboud and Karl Bringmann and Danny Hermelin and Dvir Shabtay",

note = "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: {\textcopyright} 2022 Elsevier Inc.",

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doi = "10.1016/j.jcss.2022.01.005",

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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=maxi⁡ti 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=maxi⁡ti 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.

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Scheduling lower bounds via AND subset sum

Abstract

Keywords

ASJC Scopus subject areas

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