View source for Unrelated-machines scheduling

{{Short description|Optimization problem in computer science and operations research}}
'''Unrelated-machines scheduling''' is an [[optimization problem]] in [[computer science]] and [[Operations Research|operations research]]. It is a variant of [[optimal job scheduling]]. We need to schedule ''n'' jobs ''J''<sub>1</sub>, ''J''<sub>2</sub>, ..., ''J<sub>n</sub>'' on ''m'' different machines, such that a certain objective function is optimized (usually, the [[makespan]] should be minimized). The time that machine ''i'' needs in order to process job j is denoted by ''p<sub>i,j</sub>''. The term ''unrelated'' emphasizes that there is no relation between values of ''p<sub>i,j</sub>'' for different ''i'' and ''j''. This is in contrast to two special cases of this problem: [[uniform-machines scheduling]] - in which ''p<sub>i,j</sub>''  = ''p<sub>i</sub>'' / ''s<sub>j</sub>'' (where ''s<sub>j</sub>'' is the speed of machine ''j''), and [[identical-machines scheduling]] - in which ''p<sub>i,j</sub>''  = ''p<sub>i</sub>'' (the same run-time on all machines).

In the standard [[Optimal job scheduling|three-field notation for optimal job scheduling problems]], the unrelated-machines variant is denoted by '''R''' in the first field. For example, the problem denoted by " '''R||'''<math>C_\max</math>" is an unrelated-machines scheduling problem with no constraints, where the goal is to minimize the maximum completion time. 

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 '''R||'''<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 '''R||'''<math>\sum w_i C_i</math>.

In a third variant, the goal is to ''maximize'' the ''minimum'' completion time,  " '''R||'''<math>C_\min</math>" . This variant corresponds to the problem of [[Egalitarian item allocation]].

== Algorithms ==

=== Minimizing the maximum completion time (makespan) ===
Minimizing the ''maximum'' completion time is NP-hard even for ''identical'' machines, by reduction from the [[partition problem]].

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|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|Polynomial-time approximation schemes]], 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.
Lenstra, Shmoys and Tardos<ref>{{Cite journal|last1=Lenstra|first1=Jan Karel|last2=Shmoys|first2=David B.|last3=Tardos|first3=Éva|date=1990-01-01|title=Approximation algorithms for scheduling unrelated parallel machines|url=https://doi.org/10.1007/BF01585745|journal=Mathematical Programming|language=en|volume=46|issue=1|pages=259–271|doi=10.1007/BF01585745|s2cid=52867171 |issn=1436-4646}}</ref> presented a polytime 2-factor approximation algorithm, and proved that no polytime algorithm with approximation factor smaller than 3/2 is possible unless P=NP. Closing the gap between the 2 and the 3/2 is a long-standing open problem.

Verschae and Wiese<ref>{{Cite journal|last1=Verschae|first1=José|last2=Wiese|first2=Andreas|date=2013-11-10|title=On the configuration-LP for scheduling on unrelated machines|url=https://link-springer-com.mgs.ariel.ac.il/article/10.1007/s10951-013-0359-4|journal=Journal of Scheduling|volume=17|issue=4|pages=371–383|doi=10.1007/s10951-013-0359-4|s2cid=254694965 |issn=1094-6136|arxiv=1011.4957}}</ref> presented a different 2-factor approximation algorithm.

Glass, Potts and Shade<ref>{{Cite journal|date=1994-07-01|title=Unrelated parallel machine scheduling using local search|journal=Mathematical and Computer Modelling|language=en|volume=20|issue=2|pages=41–52|doi=10.1016/0895-7177(94)90205-4|issn=0895-7177|doi-access=|last1=Glass |first1=C.A. |last2=Potts |first2=C.N. |last3=Shade |first3=P. }}</ref> compare various [[Local search (optimization)|local search]] techniques for minimizing the makespan on unrelated machines. Using computerized simulations, they find that [[tabu search]] and [[simulated annealing]] perform much better than [[Genetic algorithm|genetic algorithms]].

=== Minimizing the average completion time ===
Bruno, Coffman and Sethi<ref>{{Cite journal|last1=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|s2cid=2507557 |issn=0001-0782|doi-access=free}}</ref> present an algorithm, running in time <math>O(\max(m n^2,n^3))</math>, for minimizing the average job completion time on ''unrelated'' machines,  '''R||'''<math>\sum C_j</math> (the average over all ''jobs'', of the time it takes to complete the jobs). 

Minimizing the ''weighted average'' completion time, '''R||'''<math>\sum w_j C_j</math> (where ''w<sub>j</sub>'' is the weight of job ''j''), is NP-hard even on ''identical'' machines, by reduction from the [[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|s2cid=10956951 |issn=0004-5411|doi-access=free}}</ref>

Schulz and Skutella<ref>{{Cite journal |last1=Schulz |first1=Andreas S. |last2=Skutella |first2=Martin |date=2002-01-01 |title=Scheduling Unrelated Machines by Randomized Rounding |url=https://epubs.siam.org/doi/abs/10.1137/S0895480199357078 |journal=SIAM Journal on Discrete Mathematics |volume=15 |issue=4 |pages=450–469 |doi=10.1137/S0895480199357078 |issn=0895-4801}}</ref> present a (3/2+ε)-approximation algorithm using [[randomized rounding]]. Their algorithm is a (2+ε)-approximation for the problem with job release times, '''R|<math>r_j</math>|'''<math>\sum w_j C_j</math>.

=== Maximizing the profit ===
Bar-Noy, Bar-Yehuda, Freund, Naor and Schieber<ref>{{Cite journal|last1=Bar-Noy|first1=Amotz|last2=Bar-Yehuda|first2=Reuven|last3=Freund|first3=Ari|last4=(Seffi) Naor|first4=Joseph|last5=Schieber|first5=Baruch|author5-link=Baruch Schieber|date=2001-09-01|title=A unified approach to approximating resource allocation and scheduling|url=https://doi.org/10.1145/502102.502107|journal=Journal of the ACM|volume=48|issue=5|pages=1069–1090|doi=10.1145/502102.502107|s2cid=12329294 |issn=0004-5411}}</ref> consider a setting in which, for each job and machine, there is a ''profit'' for running this job on that machine. They present a 1/2 approximation for discrete input and (1-''ε'')/2 approximation for continuous input.

=== Maximizing the minimum completion time ===
{{Main|Egalitarian item allocation}}
Suppose that, instead of "jobs" we have valuable items, and instead of "machines" we have people. Person ''i'' values item j at ''p<sub>i,j</sub>''. We would like to allocate the items to the people, such that the least-happy person is as happy as possible. This problem is equivalent to unrelated-machines scheduling in which the goal is to maximize the minimum completion time. It is better known by the name ''egalitarian'' or ''[[max-min item allocation]]''.

== Linear programming formulation ==
A natural way to formulate the problem as a linear program is called the ''Lenstra–Shmoys–Tardos<ref name=":2">{{Cite journal|last1=Lenstra|first1=Jan Karel|last2=Shmoys|first2=David B.|last3=Tardos|first3=Éva|date=1990-01-01|title=Approximation algorithms for scheduling unrelated parallel machines|url=https://doi.org/10.1007/BF01585745|journal=Mathematical Programming|language=en|volume=46|issue=1|pages=259–271|doi=10.1007/BF01585745|issn=1436-4646|s2cid=206799898}}</ref> linear program (LST LP)''. For each machine ''i'' and job ''j<sub>,</sub>'' define a variable <math>z_{i,j}</math>, which equals 1 if machine ''i'' processes job ''j'', and 0 otherwise. Then, the LP constraints are:

* <math>\sum_{i=1}^m z_{i,j} = 1</math> for every job ''j'' in 1,...,''n;''
* <math>\sum_{j=1}^n z_{i,j}\cdot p_{i,j} \leq T</math> for every machine ''i'' in 1,...,''m;''
* <math>z_{i,j} \in \{0,1\}</math> for every ''i'', ''j''.

Relaxing the integer constraints gives a linear program with size polynomial in the input. Solving the relaxed problem can be rounded to obtain a 2-approximation to the problem.''<ref name=":2" />''

Another LP formulation is the [[configuration linear program]]. For each machine ''i'', there are finitely many subsets of jobs that can be processed by machine ''i'' in time at most ''T''. Each such subset is called a ''configuration'' for machine ''i''. Denote by ''C<sub>i</sub>''(''T'') the set of all configurations for machine ''i''. For each machine ''i'' and configuration ''c'' in ''C<sub>i</sub>''(''T''), define a variable <math>x_{i,c}</math> which equals 1 if the actual configuration used in machine ''i'' is ''c'', and 0 otherwise. Then, the LP constraints are:

* <math>\sum_{c\in C_i(T)}x_{i,c} = 1</math> for every machine ''i'' in 1,...,''m;''
* <math>\sum_{i=1}^m \sum_{c\ni j, c\in C_i(T)}x_{i,c} = 1</math> for every job ''j'' in 1,...,''n;''
* <math>x_{i,j} \in \{0,1\}</math> for every ''i'', ''j''.

Note that the number of configurations is usually exponential in the size of the problem, so the size of the configuration LP is exponential. However, in some cases it is possible to bound the number of possible configurations, and therefore find an approximate solution in polynomial time.

== Special cases ==
There is a special case in which ''p<sub>i,j</sub>''  is either 1 or infinity. In other words, each job can be processed on a subset of ''allowed machines'', and its run-time in each of these machines is 1. This variant is sometimes denoted by  " '''P|p<sub>j</sub>=1,M<sub>j</sub>|'''<math>C_\max</math>". It can be solved in polynomial time. <ref>{{Cite journal|last1=Lin|first1=Yixun|last2=Li|first2=Wenhua|date=2004-07-01|title=Parallel machine scheduling of machine-dependent jobs with unit-length|url=https://www.sciencedirect.com/science/article/pii/S0377221702009141|journal=European Journal of Operational Research|series=EURO Excellence in Practice Award 2001|language=en|volume=156|issue=1|pages=261–266|doi=10.1016/S0377-2217(02)00914-1|issn=0377-2217}}</ref>

== Extensions ==
Kim, Kim, Jang and Chen<ref>{{Cite journal|date=2002-06-01|title=Unrelated parallel machine scheduling with setup times using simulated annealing|url=https://www.sciencedirect.com/science/article/abs/pii/S0736584502000133|journal=Robotics and Computer-Integrated Manufacturing|language=en|volume=18|issue=3–4|pages=223–231|doi=10.1016/S0736-5845(02)00013-3|issn=0736-5845|last1=Kim |first1=Dong-Won |last2=Kim |first2=Kyong-Hee |last3=Jang |first3=Wooseung |last4=Frank Chen |first4=F. }}</ref> extend the problem by allowing each job to have a setup time, which depends on the job but not on the machine. They present a solution using  [[simulated annealing]]. Vallada and Ruiz<ref>{{Cite journal|date=2011-06-16|title=A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times|url=https://www.sciencedirect.com/science/article/abs/pii/S0377221711000142|journal=European Journal of Operational Research|language=en|volume=211|issue=3|pages=612–622|doi=10.1016/j.ejor.2011.01.011|issn=0377-2217|hdl-access=free|hdl=10251/35412|last1=Vallada |first1=Eva |last2=Ruiz |first2=Rubén }}</ref> present a solution using a [[genetic algorithm]].

Nisan and Ronen in their 1999 paper on [[algorithmic mechanism design]].<ref name=NisanRonen01>{{Cite journal|doi=10.1006/game.1999.0790|title=Algorithmic Mechanism Design|journal=Games and Economic Behavior|volume=35|issue=1–2|pages=166–196|year=2001|last1=Nisan|first1=Noam|last2=Ronen|first2=Amir|citeseerx=10.1.1.16.7473}}</ref> extend the problem in a different way, by assuming that the jobs are owned by selfish agents (see [[Truthful job scheduling]]).

== External links ==

* [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]]