Academic Journal
Tight Running Times for Minimum ℓ q -Norm Load Balancing: Beyond Exponential Dependencies on 1/ϵ.
| Τίτλος: | Tight Running Times for Minimum ℓ |
|---|---|
| Συγγραφείς: | Chen, Lin1 (AUTHOR) chenlin198662@zju.edu.cn, Tao, Liangde1 (AUTHOR) vast.tld@gmail.com, Verschae, José2 (AUTHOR) jverschae@uc.cl |
| Πηγή: | Mathematics of Operations Research (INFORMS). Aug2026, Vol. 51 Issue 3, p2354-2392. 39p. |
| Θεματικοί όροι: | *Scheduling, Approximation algorithms, Computational complexity, Computer scheduling, Number theory |
| Περίληψη: | We consider a classical scheduling problem on m identical machines. For an arbitrary constant q>1 , the aim is to assign jobs to machines such that ∑i=1mCiq is minimized, where Ci is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of a (1+ϵ) -approximation algorithm for a strongly NP-hard problem cannot be polynomial on 1/ϵ , unless P=NP. For most problems in the literature, this translates into algorithms with running time at least as large as 2Ω(1/ϵ)+nO(1). For the natural scheduling problem above, we establish the existence of an algorithm that violates this threshold. More precisely, we design a PTAS that runs in 2O˜(1/ϵ)+nO(1) time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between ϵ and the number of machines m. More precisely, our lower bound only holds when m=Θ(1/ϵ). Better algorithms that go beyond the lower bound exist for other values of m. In particular, there even exists an algorithm with running time polynomial in 1/ϵ if we restrict ourselves to instances with m=Ω(1/ϵlog21/ϵ). Funding: This research was supported by the Center for Mathematical Modeling BASAL [Fund FB210005], ANID-Chile, and ANID/Fondecyt [Regular 1221460]. National Science Foundation China [12271477], National Science Foundation [1756014]. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
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| Items | – Name: Title Label: Title Group: Ti Data: Tight Running Times for Minimum ℓ<subscript> q </subscript>-Norm Load Balancing: Beyond Exponential Dependencies on 1/ϵ. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Chen%2C+Lin%22">Chen, Lin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> chenlin198662@zju.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Tao%2C+Liangde%22">Tao, Liangde</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> vast.tld@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Verschae%2C+José%22">Verschae, José</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> jverschae@uc.cl</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Operations+Research+%28INFORMS%29%22">Mathematics of Operations Research (INFORMS)</searchLink>. Aug2026, Vol. 51 Issue 3, p2354-2392. 39p. – Name: Subject Label: Subject Terms Group: Su Data: *<searchLink fieldCode="DE" term="%22Scheduling%22">Scheduling</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+algorithms%22">Approximation algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+complexity%22">Computational complexity</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+scheduling%22">Computer scheduling</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider a classical scheduling problem on m identical machines. For an arbitrary constant q>1 , the aim is to assign jobs to machines such that ∑i=1mCiq is minimized, where Ci is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of a (1+ϵ) -approximation algorithm for a strongly NP-hard problem cannot be polynomial on 1/ϵ , unless P=NP. For most problems in the literature, this translates into algorithms with running time at least as large as 2Ω(1/ϵ)+nO(1). For the natural scheduling problem above, we establish the existence of an algorithm that violates this threshold. More precisely, we design a PTAS that runs in 2O˜(1/ϵ)+nO(1) time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between ϵ and the number of machines m. More precisely, our lower bound only holds when m=Θ(1/ϵ). Better algorithms that go beyond the lower bound exist for other values of m. In particular, there even exists an algorithm with running time polynomial in 1/ϵ if we restrict ourselves to instances with m=Ω(1/ϵlog21/ϵ). Funding: This research was supported by the Center for Mathematical Modeling BASAL [Fund FB210005], ANID-Chile, and ANID/Fondecyt [Regular 1221460]. National Science Foundation China [12271477], National Science Foundation [1756014]. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1287/moor.2023.0231 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 39 StartPage: 2354 Subjects: – SubjectFull: Scheduling Type: general – SubjectFull: Approximation algorithms Type: general – SubjectFull: Computational complexity Type: general – SubjectFull: Computer scheduling Type: general – SubjectFull: Number theory Type: general Titles: – TitleFull: Tight Running Times for Minimum ℓ q -Norm Load Balancing: Beyond Exponential Dependencies on 1/ϵ. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Chen, Lin – PersonEntity: Name: NameFull: Tao, Liangde – PersonEntity: Name: NameFull: Verschae, José IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 08 Text: Aug2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0364765X Numbering: – Type: volume Value: 51 – Type: issue Value: 3 Titles: – TitleFull: Mathematics of Operations Research (INFORMS) Type: main |
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