75:. In the general case, the processing time of each job may be different on different machines; in the case of identical machine scheduling, the processing time of each job is the same on each machine. Therefore, identical machine scheduling is equivalent to
1014:
1054:) is applicable when the "jobs" are actually spare parts that are required to keep the machines running, and they have different life-times. The goal is to keep machines running for as long as possible. The
906:
641:
1189:
1470:
1413:
1356:
1299:
828:
1105:
689:
1216:
1139:
759:
275:(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(
219:
314:
269:
170:
1052:
394:
122:
516:
450:
2009:
926:
1939:
469:
346:
presents an exponential-time algorithm and a polynomial-time approximation scheme for solving both these NP-hard problems on identical machines:
1744:
909:
418:
algorithm (an algorithm that processes the jobs in an arbitrary fixed order, and schedules each job to the first available machine) is a
1862:
2002:
1955:
701:
presented several approximation algorithms for any number of identical machines (even when the number of machines is not fixed):
840:
1480:, and satisfies a strong continuity assumption that they call "F*", then both minimization problems have a PTAS. Similarly, if
124:" is an identical machine scheduling problem with no constraints, where the goal is to minimize the maximum completion time.
2048:
2162:
1995:
1979:
76:
1218:
a huge constant that is exponential in the required approximation factor ε. The algorithm uses
Lenstra's algorithm for
577:
2157:
2043:
1903:
1219:
523:
68:
1144:
2033:
80:
2064:
174:
1488:, and satisfies F*, then both maximization problems have a PTAS. In both cases, the run-time of the PTAS is O(
1633:
1418:
1361:
1304:
1247:
2136:
320:
There can be many SPT schedules; finding the SPT schedule with the smallest finish time (also called OMFT –
771:
2110:
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2018:
1061:
913:
87:
72:
1818:
2105:
650:
1504:
1194:
1116:
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184:
20:
2131:
2079:
540:
28:
2115:
1799:
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535:
172:. More generally, when some jobs are more important than others, it may be desired to minimize a
549:
presented a simple polynomial-time algorithm that attains an 11/9≈1.222 approximation in time O(
289:
244:
145:
1770:"Using dual approximation algorithms for scheduling problems theoretical and practical results"
1030:
372:
336:. It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the
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24:
1899:"A polynomial-time approximation scheme for maximizing the minimum machine completion time"
421:
1485:
1477:
415:
60:
identical machines, such that a certain objective function is optimized, for example, the
1009:{\displaystyle O\left((n/\varepsilon )^{(1/\varepsilon )\log {(1/\varepsilon )}}\right)}
1649:
1916:
2151:
1834:
1754:
1671:
1629:
1055:
563:
1721:. EC '21. Budapest, Hungary: Association for Computing Machinery. pp. 630–631.
1615:
1561:
1803:
1987:
178:
of the completion time, where each job has a different weight. This is denoted by
1675:
912:. Note that, when the number of machines is a part of the input, the problem is
1715:"An Algorithmic Framework for Approximating Maximin Share Allocation of Chores"
1963:
1924:
1883:
1842:
1795:
1699:
1657:
1607:
1553:
332:
completion time is NP-hard even on identical machines, by reduction from the
1736:
1233:
consider a more general objective function. Given a positive real function
1714:
1598:
1581:
1544:
1527:
2095:
1528:"Exact and Approximate Algorithms for Scheduling Nonidentical Processors"
61:
1875:
1858:"Analysis of Greedy Solutions for a Replacement Part Sequencing Problem"
1956:
10.1002/(SICI)1099-1425(199806)1:1<55::AID-JOS2>3.0.CO;2-J
1786:
1769:
768:>0, an algorithm with approximation ratio at most (7/6+2 ) in time
709:>0, an algorithm with approximation ratio at most (6/5+2 ) in time
1691:
283:), and minimizes the average completion time on identical machines,
1938:
Alon, Noga; Azar, Yossi; Woeginger, Gerhard J.; Yadid, Tal (1998).
1727:
1719:
Proceedings of the 22nd ACM Conference on
Economics and Computation
837:>0, an algorithm with approximation ratio at most (1+ε) in time
1991:
1940:"Approximation schemes for scheduling on parallel machines"
127:
In some variants of the problem, instead of minimizing the
56:
of varying processing times, which need to be scheduled on
901:{\displaystyle O((n/\varepsilon )^{(1/\varepsilon ^{2})})}
1113:
presented a PTAS that attains an approximation factor of
88:
three-field notation for optimal job scheduling problems
473:(LPT), which sorts the jobs by descending length, is a
230:
Minimizing average and weighted-average completion time
1980:
Summary of parallel machine problems without preemtion
1421:
1364:
1307:
1250:
1197:
1147:
1119:
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1033:
929:
843:
774:
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424:
404:. Many exact and approximation algorithms are known.
375:
292:
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187:
148:
103:
543:, which has an approximation factor of 13/11≈1.182.
79:. A special case of identical machine scheduling is
2124:
2088:
2057:
2026:
1768:Hochbaum, Dorit S.; Shmoys, David B. (1987-01-01).
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67:Identical machine scheduling is a special case of
1492:), but with constants that are exponential in 1/
636:{\displaystyle O(n\cdot (n^{2}/\epsilon )^{m-1})}
358:Minimizing the maximum completion time (makespan)
1856:Friesen, D. K.; Deuermeyer, B. L. (1981-02-01).
1423:
1309:
1039:
381:
109:
574:presented a PTAS that attains (1+ε)OPT in time
131:completion time, it is desired to minimize the
90:, the identical-machines variant is denoted by
1634:"Bounds for Certain Multiprocessing Anomalies"
467:The specific list-scheduling algorithm called
2003:
1582:"Algorithms for Scheduling Independent Tasks"
1526:Horowitz, Ellis; Sahni, Sartaj (1976-04-01).
1244:, they consider the objectives of minimizing
1237:, which depends only on the completion times
8:
1676:"Bounds on Multiprocessing Timing Anomalies"
1184:{\displaystyle O(c_{\varepsilon }n\log {k})}
647:is fixed. For m=2, the run-time improves to
923:improved the run-time of this algorithm to
2010:
1996:
1988:
1819:"Bin packing with restricted piece sizes"
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1024:Maximizing the minimum completion time (
691:. The algorithm uses a technique called
1515:
1465:{\displaystyle \min _{i=1}^{m}f(C_{i})}
1408:{\displaystyle \sum _{i=1}^{m}f(C_{i})}
1351:{\displaystyle \max _{i=1}^{m}f(C_{i})}
1294:{\displaystyle \sum _{i=1}^{m}f(C_{i})}
561:), through the more general problem of
533:presented a different algorithm called
1020:Maximizing the minimum completion time
271:) can be done in polynomial time. The
1713:Huang, Xin; Lu, Pinyan (2021-07-18).
823:{\displaystyle O(n(rm^{4}+\log {n}))}
456:machines. The bound is tight for any
7:
1897:Woeginger, Gerhard J. (1997-05-01).
1575:
1573:
1571:
1521:
1519:
1100:{\displaystyle {\frac {3m-1}{4m-2}}}
71:, which is itself a special case of
1680:SIAM Journal on Applied Mathematics
135:completion time (averaged over all
94:in the first field. For example, "
1863:Mathematics of Operations Research
1650:10.1002/j.1538-7305.1966.tb01709.x
684:{\displaystyle O(n^{2}/\epsilon )}
14:
1817:Leung, Joseph Y-T. (1989-05-08).
353:Weighted-average-completion-time.
1211:{\displaystyle c_{\varepsilon }}
1134:{\displaystyle 1-{\varepsilon }}
754:{\displaystyle O(n(r+\log {n}))}
460:. This algorithm runs in time O(
400:machines, by reduction from the
350:Optimal average-completion-time;
1580:Sahni, Sartaj K. (1976-01-01).
1231:Alon, Azar, Woeginger and Yadid
214:{\displaystyle \sum w_{i}C_{i}}
1823:Information Processing Letters
1459:
1446:
1402:
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584:
1:
1917:10.1016/S0167-6377(96)00055-7
1638:Bell System Technical Journal
470:Longest Processing Time First
17:Identical-machines scheduling
1835:10.1016/0020-0190(89)90223-8
522:machines. It is also called
77:multiway number partitioning
1904:Operations Research Letters
1226:General objective functions
2179:
1220:integer linear programming
916:, so no FPTAS is possible.
531:Coffman, Garey and Johnson
524:greedy number partitioning
309:{\displaystyle \sum C_{i}}
264:{\displaystyle \sum C_{i}}
165:{\displaystyle \sum C_{i}}
69:uniform machine scheduling
1047:{\displaystyle C_{\min }}
389:{\displaystyle C_{\max }}
117:{\displaystyle C_{\max }}
81:single-machine scheduling
539:, using techniques from
511:{\displaystyle 4/3-1/3m}
322:optimal mean finish time
139:jobs); it is denoted by
2137:Truthful job scheduling
2089:Optimization objectives
1737:10.1145/3465456.3467555
2019:Optimal job scheduling
1472:. They prove that, if
1466:
1442:
1409:
1385:
1352:
1328:
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1212:
1185:
1135:
1101:
1048:
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824:
755:
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637:
512:
446:
396:) is NP-hard even for
390:
310:
265:
215:
166:
118:
73:optimal job scheduling
1944:Journal of Scheduling
1599:10.1145/321921.321934
1545:10.1145/321941.321951
1467:
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1308:
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1136:
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1049:
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693:interval partitioning
686:
638:
513:
447:
445:{\displaystyle 2-1/m}
391:
311:
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216:
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119:
1419:
1362:
1305:
1248:
1195:
1145:
1117:
1062:
1031:
927:
841:
772:
713:
651:
643:. It is an FPTAS if
578:
566:allocation of chores
477:
422:
373:
290:
245:
185:
146:
101:
21:optimization problem
2163:Number partitioning
2132:Interval scheduling
1876:10.1287/moor.6.1.74
699:Hochbaum and Shmoys
29:operations research
2158:Optimal scheduling
2125:Other requirements
2049:Unrelated machines
2039:Identical machines
1774:Journal of the ACM
1586:Journal of the ACM
1532:Journal of the ACM
1505:Fernandez's method
1462:
1405:
1348:
1291:
1208:
1181:
1131:
1097:
1044:
1006:
898:
820:
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536:multifit algorithm
518:approximation for
508:
452:approximation for
442:
386:
306:
261:
211:
162:
114:
2145:
2144:
1787:10.1145/7531.7535
1746:978-1-4503-8554-1
1484:is non-negative,
1476:is non-negative,
1415:, and maximizing
1107:of the optimum.
1095:
1058:attains at least
402:partition problem
366:completion time (
338:partition problem
238:completion time (
2170:
2058:Multi-stage jobs
2044:Uniform machines
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1644:(9): 1563–1581.
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948:
914:strongly NP-hard
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334:knapsack problem
330:weighted average
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175:weighted average
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123:
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86:In the standard
25:computer science
2178:
2177:
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2016:
1985:
1976:
1971:
1937:
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1932:
1896:
1895:
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1816:
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1811:
1767:
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1712:
1711:
1707:
1692:10.1137/0117039
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1628:
1627:
1623:
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416:list scheduling
376:
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362:Minimizing the
360:
328:Minimizing the
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234:Minimizing the
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64:is minimized.
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48:
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31:. We are given
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2065:Parallel tasks
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2034:Single machine
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2027:One-stage jobs
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1974:External links
1972:
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1911:(4): 149–154.
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1829:(3): 145–149.
1809:
1780:(1): 144–162.
1760:
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1705:
1686:(2): 416–429.
1674:(1969-03-01).
1672:Graham, Ron L.
1663:
1630:Graham, Ron L.
1621:
1592:(1): 116–127.
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1538:(2): 317–327.
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1056:LPT algorithm
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547:Huang and Lu
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541:bin packing
2152:Categories
2116:Throughput
1728:1907.04505
1511:References
225:Algorithms
2111:Tardiness
2101:Earliness
2075:Flow shop
2070:Open shop
1964:1099-1425
1925:0167-6377
1884:0364-765X
1843:0020-0190
1796:0004-5411
1755:195874333
1700:0036-1399
1658:1538-7305
1608:0004-5411
1554:0004-5411
1367:∑
1253:∑
1204:ε
1171:
1160:ε
1128:ε
1124:−
1111:Woeginger
1089:−
1075:−
993:ε
978:
969:ε
950:ε
882:ε
862:ε
807:
738:
676:ϵ
623:−
612:ϵ
591:⋅
520:identical
492:−
454:identical
429:−
398:identical
294:∑
249:∑
189:∑
150:∑
2106:Lateness
2096:Makespan
2080:Job shop
2021:problems
1632:(1966).
1616:10956951
1562:18693114
1499:See also
1191:, where
1141:in time
833:For any
764:For any
705:For any
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1804:9739129
1486:concave
364:maximum
236:average
133:average
129:maximum
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408:Graham
19:is an
1800:S2CID
1751:S2CID
1723:arXiv
1612:S2CID
1558:S2CID
921:Leung
572:Sahni
343:Sahni
35:jobs
1960:ISSN
1921:ISSN
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1839:ISSN
1792:ISSN
1741:ISBN
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1550:ISSN
910:PTAS
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414:Any
279:log
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1952:doi
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