Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s ≥ s ≥ s ≥ s , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incomp...
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| Vydáno v: | Bulletin of the Polish Academy of Sciences. Technical sciences Ročník 65; číslo 1; s. 29 - 34 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Warsaw
De Gruyter Open
01.02.2017
Polish Academy of Sciences |
| Témata: | |
| ISSN: | 2300-1917, 0239-7528, 2300-1917 |
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| Abstract | In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s
≥ s
≥ s
≥ s
, respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s
= s
= s
. If, however, Δ ≤ 4 and s
≥ 12s
, s
= s
= s
, then the problem can be solved to optimality in time O(n
). The same algorithm returns a solution of value at most 2 times optimal provided that s
≥ 2s
. Finally, we study the case s
≥ s
≥ s
= s
and give a 32/15-approximation algorithm running also in O(n
) time. |
|---|---|
| AbstractList | In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s[1] ≥ s[2] ≥ s[3] ≥ s[4], respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s[1] = s[2] = s[3]. If, however, Δ ≤ 4 and s[1] ≥ 12s[2], s[2] = s[3] = s[4], then the problem can be solved to optimality in time O(n[1.5]). The same algorithm returns a solution of value at most 2 times optimal provided that s[1] ≥ 2s[2]. Finally, we study the case s[1] ≥ s[2] ≥ s[3] = s[4] and give a 32/15-approximation algorithm running also in O(n[1.5]) time. In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s ≥ s ≥ s ≥ s , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s = s = s . If, however, Δ ≤ 4 and s ≥ 12s , s = s = s , then the problem can be solved to optimality in time O(n ). The same algorithm returns a solution of value at most 2 times optimal provided that s ≥ 2s . Finally, we study the case s ≥ s ≥ s = s and give a 32/15-approximation algorithm running also in O(n ) time. In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s 1 ≥ s 2 ≥ s 3 ≥ s 4 , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s 1 = s 2 = s 3 . If, however, Δ ≤ 4 and s 1 ≥ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5 ). The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ≥ 2s 2 . Finally, we study the case s 1 ≥ s 2 ≥ s 3 = s 4 and give a 32/15-approximation algorithm running also in O(n 1.5 ) time. |
| Author | Furmańczyk, H. Kubale, M. |
| Author_xml | – sequence: 1 givenname: H. surname: Furmańczyk fullname: Furmańczyk, H. email: hanna@inf.ug.edu.pl organization: Institute of Informatics, University of Gdańsk, Wita Stwosza 57, 80-309 Gdańsk, Poland – sequence: 2 givenname: M. surname: Kubale fullname: Kubale, M. organization: Department of Algorithms and System Modelling, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland |
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| References | Furmańczyk (ref51) 2016 Kubale (ref111) 1989 Chen (ref71) 2012 Li (ref61) 2014 Boudhar (ref11) 2005 Hopcroft (ref81) 1973; 5 Finke (ref21) 2008 Boudhar (ref01) 2003 Werra (ref41) 2005 |
| References_xml | – start-page: 312 year: 2012 ident: ref71 article-title: Equitable Δ - coloring of graphs publication-title: Disc Math – start-page: 74 year: 1989 ident: ref111 article-title: Interval vertex - coloring of a graph with forbidden colors publication-title: Disc Math – start-page: 146 year: 2005 ident: ref41 article-title: de A hypocoloring model for batch scheduling publication-title: Disc Appl Math – start-page: 156 year: 2008 ident: ref21 article-title: Batch processing with interval graph compatibilities between tasks publication-title: Disc Appl Math – start-page: 57 year: 2003 ident: ref01 article-title: Scheduling a batch processing machine with bipartite compatibility graphs publication-title: Math Methods Oper Res – start-page: 114 year: 2014 ident: ref61 article-title: Serial batch scheduling on uniform parallel machines to minimize total completion time publication-title: Process Lett – volume: 5 start-page: 2 year: 1973 ident: ref81 article-title: An algorithm for maximum matchings in bipartite graphs publication-title: SIAM J Comput – year: 2016 ident: ref51 article-title: Scheduling of unit - length jobs with cubic Incompatibility graphs on three uniform machines in print available online Feb publication-title: Disc Appl Math doi: 10.1016/j.dam.2016.01.036 – start-page: 391 year: 2005 ident: ref11 article-title: Scheduling on a batch processing machine with split compatibility graphs publication-title: Math Modell Algorithms doi: 10.1007/s10852-005-3083-z |
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| Snippet | In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s
≥ s
≥ s
≥ s
, respectively. Our aim is to find a... In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s 1 ≥ s 2 ≥ s 3 ≥ s 4 , respectively. Our aim is to find... In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s[1] ≥ s[2] ≥ s[3] ≥ s[4], respectively. Our aim is to... |
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| SubjectTerms | equitable coloring Incompatibility NP-hardness Optimization polynomial algorithm Scheduling uniform machine |
| Title | Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines |
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