Uniform-machines scheduling: Difference between revisions

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{{Short description|Optimization prpblem}}

'''Uniform machine scheduling''' (also called '''uniformly-related machine scheduling''' or '''related machine scheduling''') is an [[optimization problem]] in [[computer science]] and [[Operations Research|operations research]]. It is a variant of [[optimal job scheduling]]. We are given ''n'' jobs ''J''₁, ''J''₂, ..., ''J_n'' of varying processing times, which need to be scheduled on ''m'' different machines. The goal is to minimize the [[makespan]] - the total time required to execute the schedule. The time that machine ''i'' needs in order to process job j is denoted by ''p_i,j''. In the general case, the times ''p_i,j'' are unrelated, and any matrix of positive processing times is possible. In the specific variant called ''uniform machine scheduling'', some machines are ''uniformly'' faster than others. This means that, for each machine ''i'', there is a speed factor ''s_i'', and the run-time of job ''j'' on machine ''i'' is ''p_i,j'' = ''p_j'' / ''s_i''.

In the standard [[Optimal job scheduling|three-field notation for optimal job scheduling problems]], the uniform-machine variant is denoted by '''Q''' in the first field. For example, the problem denoted by " '''Q||'''<math>C_\max</math>" is a uniform machine scheduling problem with no constraints, where the goal is to minimize the maximum completion time. A special case of uniform machine scheduling is [[identical machine-machines scheduling]], in which all machines have the same speed. This variant is denoted by '''P''' in the first field.

In some variants of the problem, instead of minimizing the ''maximum'' completion time, it is desired to minimize the ''average'' completion time (averaged over all ''n'' jobs); it is denoted by '''Q||'''<math>\sum C_i</math>. More generally, when some jobs are more important than others, it may be desired to minimize a ''[[Weighted arithmetic mean|weighted average]]'' of the completion time, where each job has a different weight. This is denoted by '''Q||'''<math>\sum w_i C_i</math>.

== Algorithms ==

=== Minimizing the average completion time ===

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A constant-factor approximation is attained by the [[Longest-processing-time-first scheduling#uniform|Longest-processing-time-first algorithm]] (LPT).

'''Horowitz and Sahni<ref name=":0">{{Cite journal|last1=Horowitz|first1=Ellis|last2=Sahni|first2=Sartaj|date=1976-04-01|title=Exact and Approximate Algorithms for Scheduling Nonidentical Processors|url=https://doi.org/10.1145/321941.321951|journal=Journal of the ACM|volume=23|issue=2|pages=317–327|doi=10.1145/321941.321951|s2cid=18693114 |issn=0004-5411|doi-access=free}}</ref>''' presented:

* Exact [[dynamic programming]] algorithms for minimizing the ''maximum'' completion time on both uniform and unrelated machines. These algorithms run in exponential time (recall that these problems are all NP-hard).

* [[Polynomial-time approximation scheme]]s, which for any ''ε''>0, attain at most (1+ε)OPT. For minimizing the ''maximum'' completion time on two ''uniform'' machines, their algorithm runs in time <math>O(10^{2l} n)</math>, where <math>l</math> is the smallest integer for which <math>\epsilon \geq 2\cdot 10^{-l}</math>. Therefore, the run-time is in <math>O( n / \epsilon^2)</math>, so it is an [[FPTAS]]. For minimizing the ''maximum'' completion time on two ''unrelated'' machines, the run-time is <math>O(10^{l} n^2)</math> = <math>O( n^2 / \epsilon)</math>. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case.

'''Hochbaum and Shmoys'''<ref name=":02">{{Cite journal|last1=Hochbaum|first1=Dorit S.|last2=Shmoys|first2=David B.|date=1987-01-01|title=Using dual approximation algorithms for scheduling problems theoretical and practical results|journal=Journal of the ACM|volume=34|issue=1|pages=144–162|doi=10.1145/7531.7535|issn=0004-5411|s2cid=9739129|doi-access=free}}</ref> presented several approximation algorithms for any number of [[Identical-machines scheduling|''identical'' machines]]. Later,<ref>{{Cite journal|last1=Hochbaum|first1=Dorit S.|last2=Shmoys|first2=David B.|date=1988-06-01|title=A Polynomial Approximation Scheme for Scheduling on Uniform Processors: Using the Dual Approximation Approach|url=https://epubs.siam.org/doi/abs/10.1137/0217033|journal=SIAM Journal on Computing|volume=17|issue=3|pages=539–551|doi=10.1137/0217033|issn=0097-5397|url-access=subscription}}</ref> they developed a PTAS for ''uniform'' machines.

'''Epstein and Sgall'''<ref>{{Cite journal|last1=Epstein|first1=Leah|last2=Sgall|first2=Jiri|date=2004-05-01|title=Approximation Schemes for Schedulingon Uniformly Related and Identical Parallel Machines|url=https://doi.org/10.1007/s00453-003-1077-7|journal=Algorithmica|language=en|volume=39|issue=1|pages=43–57|doi=10.1007/s00453-003-1077-7|s2cid=12965369 |issn=1432-0541|url-access=subscription}}</ref> generalized the PTAS for uniform machines to handle more general objective functions. Let ''C_i'' (for ''i'' between 1 and ''m'') be the makespan of machine ''i'' in a given schedule. Instead of minimizing the objective function max(''C_i''), one can minimize the objective function max(''f''(''C_i'')), where ''f'' is any fixed function. Similarly, one can minimize the objective function sum(''f''(''C_i'')).

=== Monotonicity and Truthfulness ===

In some settings, the machine speed is the machine's private information, and we want to incentivize machines to reveal their true speed, that is, we want a [[truthful mechanism]]. An important consideration for attaining truthfulness is ''monotonicity''.<ref>{{Cite book|last1=Archer|first1=A.|last2=Tardos|first2=E.|title=Proceedings 42nd IEEE Symposium on Foundations of Computer Science |chapter=Truthful mechanisms for one-parameter agents |date=2001-10-01|chapter-url=https://ieeexplore.ieee.org/document/959924|pages=482–491|doi=10.1109/SFCS.2001.959924|isbn=0-7695-1390-5 |s2cid=11377808 }}</ref> It means that, if a machine reports a higher speed, and all other inputs remain the same, then the total processing time allocated to the machine weakly increases. For this problem:

* '''Auletta, De Prisco, Penna and Persiano'''<ref>{{Cite journalbook|last1=Auletta|first1=Vincenzo|last2=De Prisco|first2=Roberto|last3=Penna|first3=Paolo|last4=Persiano|first4=Giuseppe|title=Stacs 2004 |chapter=Deterministic Truthful Approximation Mechanisms for Scheduling Related Machines |date=2004|editor-last=Diekert|editor-first=Volker|editor2-last=Habib|editor2-first=Michel|title=Deterministic Truthful Approximation Mechanisms for Scheduling Related Machines|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-24749-4_53|journal=Stacs 2004|series=Lecture Notes in Computer Science|volume=2996 |language=en|___location=Berlin, Heidelberg|publisher=Springer|pages=608–619|doi=10.1007/978-3-540-24749-4_53|isbn=978-3-540-24749-4}}</ref> presented a 4-approximation monotone algorithm, which runs in polytime when the number of machines is fixed.

*'''Ambrosio and Auletta'''<ref>{{Cite journalbook|last1=Ambrosio|first1=Pasquale|last2=Auletta|first2=Vincenzo|title=Approximation and Online Algorithms |chapter=Deterministic Monotone Algorithms for Scheduling on Related Machines |date=2005|editor-last=Persiano|editor-first=Giuseppe|editor2-last=Solis-Oba|editor2-first=Roberto|title=Deterministic Monotone Algorithms for Scheduling on Related Machines|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-31833-0_22|journal=Approximation and Online Algorithms|series=Lecture Notes in Computer Science|volume=3351 |language=en|___location=Berlin, Heidelberg|publisher=Springer|pages=267–280|doi=10.1007/978-3-540-31833-0_22|isbn=978-3-540-31833-0}}</ref> proved that the [[Longest processing time|Longest Processing Time]] algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78. In contrast, [[List scheduling]] is not monotone for ''c'' > 2.

* '''Andelman, Azar and Sorani'''<ref>{{Cite journalbook|last1=Andelman|first1=Nir|last2=Azar|first2=Yossi|last3=Sorani|first3=Motti|title=Stacs 2005 |chapter=Truthful Approximation Mechanisms for Scheduling Selfish Related Machines |date=2005|editor-last=Diekert|editor-first=Volker|editor2-last=Durand|editor2-first=Bruno|title=Truthful Approximation Mechanisms for Scheduling Selfish Related Machines|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-31856-9_6|journal=Stacs 2005|series=Lecture Notes in Computer Science|volume=3404 |language=en|___location=Berlin, Heidelberg|publisher=Springer|pages=69–82|doi=10.1007/978-3-540-31856-9_6|isbn=978-3-540-31856-9}}</ref> presented a 5-approximation monotone algorithm, which runs in polytime even when the number of machines is variable.

* '''Kovacz'''<ref>{{Cite book|last=Kovács|first=Annamária|title=Algorithms – ESA 2005 |chapter=Fast Monotone 3-Approximation Algorithm for Scheduling Related Machines |date=2005|editor-last=Brodal|editor-first=Gerth Stølting|editor2-last=Leonardi|editor2-first=Stefano |chapter-url=https://link.springer.com/chapter/10.1007/11561071_55 |series=Lecture Notes in Computer Science|volume=3669 |language=en|___location=Berlin, Heidelberg|publisher=Springer|pages=616–627|doi=10.1007/11561071_55|isbn=978-3-540-31951-1}}</ref> presented a 3-approximation monotone algorithm.

== Extensions ==

'''Dependent jobs''': In some cases, the jobs may be dependent. For example, take the case of reading user credentials from console, then use it to authenticate, then if authentication is successful display some data on the console. Clearly one task is dependent upon another. This is a clear case of where some kind of [[Orderorder theory|ordering]] exists between the tasks. In fact it is clear that it can be modelled with [[partial ordering]]. Then, by definition, the set of tasks constitute a [[Latticelattice (order)|lattice structure]]. This adds further complication to the multiprocessor scheduling problem.

'''Static versus Dynamic''': Machine scheduling algorithms are static or dynamic. A scheduling algorithm is '''static''' if the scheduling decisions as to what computational tasks will be allocated to what processors are made before running the program. An algorithm is '''dynamic''' if it is taken at run time. For static scheduling algorithms, a typical approach is to rank the tasks according to their precedence relationships and use a list scheduling technique to schedule them onto the processors.<ref>{{Cite journal|last1=Kwok|first1=Yu-Kwong|last2=Ahmad|first2=Ishfaq|date=1999-12-01|title=Static scheduling algorithms for allocating directed task graphs to multiprocessors|journal=ACM Computing Surveys|volume=31|issue=4|pages=406–471|doi=10.1145/344588.344618|issn=0360-0300|citeseerx=10.1.1.322.2295|s2cid=207614150 }}</ref>

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== External links ==

* [httphttps://www2.informatik.uni-osnabrueck.de/knust/class/dateien/classes/pqr_nop/pqr_nop/ Summary of parallel machine problems without preemtion]

* [http://www2.informatik.uni-osnabrueck.de/knust/class/dateien/classes/pqr_nop/pqr_nop/ Summary of parallel machine problems without preemtion]

==References==

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[[Category:Optimal scheduling]]

[[Category:NP-complete problems]]