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-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>.

A special case of uniform machine scheduling is '''identical machine scheduling''', in which all machines have the same speed. This variant is denoted by '''P''' in the first field. For example, " '''P||'''<math>C_\max</math>" is an identical machine scheduling problem with no constraints, where the goal is to minimize the maximum completion time. This problem is also a special case of the [[multiway number partitioning]] problem. In both problems, the goal is to partition numbers into subsets with nearly-equal sum; identical machine scheduling is a special case in which the objective is to minimize the largest sum. Many algorithms for multiway number partitioning, both exact and approximate, can be used to attain this objective.

== Algorithms ==

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 ===

Minimizing the ''average'' completion time can be done in polynomial time:

* The '''SPT algorithm''' (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far. It runs in time O(''n'' log ''n''), and minimizes the average completion time on ''identical'' machines,<ref>{{Cite journal|lastname=Horowitz|first=Ellis|last2=Sahni|first2=Sartaj|date=1976-04-01|title=Exact and Approximate Algorithms for Scheduling Nonidentical Processors|url=https":0"//doi.org/10.1145/321941.321951|journal=Journal of the ACM|volume=23|issue=2|pages=317–327|doi=10.1145/321941.321951|issn=0004-5411}}</ref> '''P||'''<math>\sum C_i</math>.

*'''Horowitz and Sahni<ref name=":0" />''' present an exact algorithm, with run time O(''n'' log ''m n''), for minimizing the average completion time on ''uniform'' machines, '''Q||'''<math>\sum C_i</math>.

**There can be many SPT schedules; finding the SPT schedule with the smallest finish time (also called OMFT - optimal mean finish time) is NP-hard. Sahni<ref name=":3" /> presents an exponential-time algorithm and a polynomial-time approximation algorithm for identical machines.

* '''HorowitzBruno, Coffman and SahniSethi'''<ref>{{Cite namejournal|last1=":0"Bruno|first1=J.|last2=Coffman|first2=E. G.|last3=Sethi|first3=R.|date=1974-07-01|title=Scheduling independent tasks to reduce mean finishing time|journal=Communications of the ACM|volume=17|issue=7|pages=382–387|doi=10.1145/361011.361064|issn=0001-0782|doi-access=free}}</ref>''' present an exact algorithm, withrunning runin time <math>O(''n'' log ''\max(m n''^2,n^3))</math>, for minimizing the average completion time on ''uniformunrelated'' machines, '''QR||'''<math>\sum C_i</math>.

* '''Bruno, Coffman and Sethi'''<ref>{{Cite journal|last=Bruno|first=J.|last2=Coffman|first2=E. G.|last3=Sethi|first3=R.|date=1974-07-01|title=Scheduling independent tasks to reduce mean finishing time|url=https://doi.org/10.1145/361011.361064|journal=Communications of the ACM|volume=17|issue=7|pages=382–387|doi=10.1145/361011.361064|issn=0001-0782}}</ref> present an algorithm, running in time <math>O(\max(m n^2,n^3))</math>, for minimizing the average completion time on ''unrelated'' machines, '''R||'''<math>\sum C_i</math>.

=== Minimizing the weighted-average completion time ===

Minimizing the ''weighted average'' completion time is NP-hard even on ''identical'' machines, by reduction from the [[Knapsack problem|knapsack problem.]].'''<ref name=":0" />''' It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the [[partition problem]].<ref name=":3">{{Cite journal|last=Sahni|first=Sartaj K.|date=1976-01-01|title=Algorithms for Scheduling Independent Tasks|journal=Journal of the ACM|volume=23|issue=1|pages=116–127|doi=10.1145/321921.321934|issn=0004-5411|doi-access=free}}</ref>

'''Sahni<ref name=":3" />''' presents an exponential-time algorithm and a polynomial-time approximation algorithm for ''identical'' machines.

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* Exact [[dynamic programming]] algorithms for minimizing the ''weighted-average completion time'' on ''uniform'' machines. These algorithms run in exponential time.

* [[Polynomial-time approximation scheme]]<nowiki/>s, which for any ''ε''>0, attain at most (1+ε)OPT. For minimizing the ''weighted average'' completion time on two ''uniform'' machines, the run-time is <math>O(10^{l} n^2)</math> = <math>O( n^2 / \epsilon)</math>, so it is an FPTAS. 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. They do not present an algorithm for ''weighted-average'' completion time on ''unrelated'' machines.

=== Minimizing the maximum completion time (makespan) ===

Minimizing the ''maximum'' completion time is NP-hard even for ''identical'' machines, by reduction from the [[partition problem]]. Many approximation algorithms are known.

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

'''Graham''' proved that:

'''Horowitz and Sahni<ref name=":0">{{Cite journal|lastlast1=Horowitz|firstfirst1=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:

* Any [[list scheduling]] algorithm (an algorithm that processes the jobs in an arbitrary fixed order, and schedules each job to the first available machine) is a <math>2-1/m</math> approximation for ''identical'' machines.<ref name=":2">{{Cite journal|last=Graham|first=Ron L.|author-link=Ronald Graham|date=1966|title=Bounds for Certain Multiprocessing Anomalies|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1966.tb01709.x|journal=Bell System Technical Journal|language=en|volume=45|issue=9|pages=1563–1581|doi=10.1002/j.1538-7305.1966.tb01709.x|issn=1538-7305}}</ref> The bound is tight for any ''m''. This algorithm runs in time O(''n'').

* The specific list-scheduling algorithm called '''Longest Processing Time First''' (LPT), which sorts the jobs by descending length, is a <math>4/3-1/3m</math> approximation for ''identical'' machines.<ref name=":1">{{Cite journal|last=Graham|first=Ron L.|author-link=Ronald Graham|date=1969-03-01|title=Bounds on Multiprocessing Timing Anomalies|url=https://epubs.siam.org/doi/abs/10.1137/0117039|journal=SIAM Journal on Applied Mathematics|volume=17|issue=2|pages=416–429|doi=10.1137/0117039|issn=0036-1399}}</ref>{{Rp|sec.5}} It has good asymptotic convergence properties even for ''uniform'' machines.<ref>{{Cite journal|last=Frenk|first=J. B. G.|last2=Rinnooy Kan|first2=A. H. G.|date=1987-05-01|title=The Asymptotic Optimality of the LPT Rule|url=https://pubsonline.informs.org/doi/abs/10.1287/moor.12.2.241|journal=Mathematics of Operations Research|volume=12|issue=2|pages=241–254|doi=10.1287/moor.12.2.241|issn=0364-765X}}</ref> It is also called [[greedy number partitioning]].

'''Sahni'''<ref name=":3">{{Cite journal|last=Sahni|first=Sartaj K.|date=1976-01-01|title=Algorithms for Scheduling Independent Tasks|url=https://doi.org/10.1145/321921.321934|journal=Journal of the ACM|volume=23|issue=1|pages=116–127|doi=10.1145/321921.321934|issn=0004-5411}}</ref> presented several algorithms for ''identical'' machines. In particular:

* A PTAS that attains (1+ε)OPT in time <math>O(n\cdot (n^2 / \epsilon)^{m-1})</math>. It is an FPTAS if ''m'' is fixed. For m=2, the run-time improves to <math>O(n^2 / \epsilon)</math>. The algorithm uses a technique called ''interval partitioning''.

'''Horowitz and Sahni<ref name=":0">{{Cite journal|last=Horowitz|first=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|issn=0004-5411}}</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]]<nowiki/>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|url=https://doi.org/10.1145/7531.7535|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]]. (evenLater,<ref>{{Cite whenjournal|last1=Hochbaum|first1=Dorit theS.|last2=Shmoys|first2=David numberB.|date=1988-06-01|title=A ofPolynomial machinesApproximation isScheme notfor fixed)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|lastlast1=Epstein|firstfirst1=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'')).

* For any ''r'' >0, an algorithm with approximation ratio at most (6/5+2^−''r'' ) in time <math>O(n(r+\log{n}))</math>.

* For any ''r'' >0, an algorithm with approximation ratio at most (7/6+2^−''r'' ) in time <math>O(n(r m^4+\log{n}))</math>.

* For any ''ε''>0, an algorithm with approximation ratio at most (1+ε) in time <math>O((n/\varepsilon)^{(1/\varepsilon^2)})</math> . This is a [[Polynomial-time approximation scheme|PTAS]]. Note that, when the number of machines is a part of the input, the problem is [[Strong NP-completeness|strongly NP-hard]], so no FPTAS is possible.

**The run-time of this algorithm was later improved to <math>O\left((n/\varepsilon)^{(1/\varepsilon)\log{(1/\varepsilon)}}\right)</math>.<ref>{{Cite journal|date=1989-05-08|title=Bin packing with restricted piece sizes|url=https://www.sciencedirect.com/science/article/abs/pii/0020019089902238|journal=Information Processing Letters|language=en|volume=31|issue=3|pages=145–149|doi=10.1016/0020-0190(89)90223-8|issn=0020-0190}}</ref>

=== Monotonicity and Truthfulness ===

Later,<ref>{{Cite journal|last=Hochbaum|first=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}}</ref> they developed a PTAS for ''uniform'' machines.

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|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 book|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|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-24749-4_53|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.

'''Epstein and Sgall'''<ref>{{Cite journal|last=Epstein|first=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|issn=1432-0541}}</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'')).

*'''Ambrosio and Auletta'''<ref>{{Cite book|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|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-31833-0_22|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 book|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|chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-31856-9_6|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|lastlast1=Kwok|firstfirst1=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>

'''Multi-stage jobs''': In various settings, each job might have several operations that must be executed in parallel. Some such settings are handled by [[Open-shop scheduling|open shop scheduling]], [[flow shop scheduling]] and [[job shop scheduling]].

== 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==

<references/>

*

{{Scheduling problems}}

[[Category:Optimal scheduling]]

[[Category:NumberNP-complete partitioningproblems]]