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## Is A Unit-Time Job Shop Not Easier Than Identical Parallel Machines? (1998)

Venue: | Discrete Appl. Math |

Citations: | 8 - 7 self |

### Citations

13824 |
Computers and Intractability - A Guide to the Theory of NPCompleteness, Bell Telephone Laboratories, Incorporated
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...notations of one-operation job processing times. The pair (I; k) will denote the instance of the decision problem related to an instance I of the scheduling problem jjswith upper bound k forsvalues [G=-=J79]-=-. We consider only decision problems belonging to NP [GJ79], so we will assume that for any instance (I; k) there exists a feasible schedule having a code of length l polynomial in size of (I; k). Thu... |

1934 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ... 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC max / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2kw j C j / J2jno wait; p ij = 1jw j C j Table 5. Reduc... |

324 | Sequencing and scheduling: Algorithms and complexity
- Lawler, Lenstra, et al.
- 1993
(Show Context)
Citation Context ...ij = 1jC max / P2jjC max J2jp ij = 1jC max / P2jpmtnjC max In what follow, we use the well-known three-eld classication jjsfor scheduling problems described in the surveys [GLLRK79], [LLRK82] and [LLR=-=KS93-=-], where ,sandsspecify the machine environment, job characteristics and the minimization criterion, respectively. The above two reductions allow to suggest that under the unit-processing-time constrai... |

304 |
Theory of Scheduling
- Conway, Maxwell, et al.
- 1967
(Show Context)
Citation Context ...1js; P 2 js3 5 js/ J 2 jno wait;s3 5 ; p ij = 1js: In the following evident corollary from Theorem 3.1 we take into account the equivalences P 2 jpmtnjw j C j P 2 kw j C j [M59], 1jpmtnjs 1ks[CMM67] and the trivial reduction 1jjs/ P2jj . Corollary 3.1. The reductions listed below prove the NP-hardness or strong NP-hardness of unit-time job-shop scheduling problems listed in the third column. T... |

249 |
Complexity of machine scheduling problems
- LENSTRA, KAN, et al.
- 1977
(Show Context)
Citation Context ...nces to which are listed in thesrst column. Table 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC max / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2kw j C j ... |

158 |
Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: A survey
- Graham, Lawler, et al.
- 1979
(Show Context)
Citation Context ...ductions: J2jno wait; p ij = 1jC max / P2jjC max J2jp ij = 1jC max / P2jpmtnjC max In what follow, we use the well-known three-eld classication jjsfor scheduling problems described in the surveys [GLL=-=RK79-=-], [LLRK82] and [LLRKS93], where ,sandsspecify the machine environment, job characteristics and the minimization criterion, respectively. The above two reductions allow to suggest that under the unit-... |

118 |
Scheduling with deadlines and loss functions
- McNaughton
- 1959
(Show Context)
Citation Context ...J 2 js3 5 ; p ij = 1js; P 2 js3 5 js/ J 2 jno wait;s3 5 ; p ij = 1js: In the following evident corollary from Theorem 3.1 we take into account the equivalences P 2 jpmtnjw j C j P 2 kw j C j [M59], 1jpmtnjs 1ks[CMM67] and the trivial reduction 1jjs/ P2jj . Corollary 3.1. The reductions listed below prove the NP-hardness or strong NP-hardness of unit-time job-shop scheduling problems listed ... |

112 |
strong”np-completeness results: Motivation, examples, and implications
- Garey, Johnson
- 1978
(Show Context)
Citation Context ...As Lenstra [L] points out, the NP-hardness proofs in the strong sense for J2jchains; p ij = 1jC j , J2jr j ; p ij = 1jw j C j and J2jp ij = 1jw j T j can be obtained by reductions from 3-PARTITION [GJ78]. A UNIT-TIME JOB SHOP 11 Concluding remarks Summing up we can infer that the complexity status of unit-time job-shop problems with criteria C max , C j , w j C j , T j , w j T j , U j and w j U... |

100 |
Scheduling independent tasks to reduce mean finishing time
- Coffman, Sethi
- 1974
(Show Context)
Citation Context ...are listed in the first column. Table 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j j\SigmaC j / J2jr j ; p ij = 1j\SigmaC j [DLW92] P2jpmtn; r j j\SigmaU j / J2jr j ; p ij = 1j\SigmaU j =-=[BCS74]-=-,[LRKB77] P2jpmtnj\Sigmaw j C j / J2jp ij = 1j\Sigmaw j C j [K72] 1jpmtnj\Sigmaw j U j / J2jp ij = 1j\Sigmaw j U j [DL90] 1jpmtnj\SigmaT j / J2jp ij = 1j\SigmaT j [LRKB77] P2kC max / J2jno wait; p ij ... |

77 |
Minimizing total tardiness on one machine is np-hard
- Du, Leung
- 1990
(Show Context)
Citation Context ...] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC max / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2kw j C j / J2jno wait; p ij = 1jw j C j Table 5. Reductions proving the strong NP-hardness [DLY91] P... |

70 |
A 'pseudopolynomial' algorithm for sequencing jobs to minimize total tardiness
- Lawler
- 1977
(Show Context)
Citation Context .../ J2jno wait; p ij = 1jw j C j Table 5. Reductions proving the strong NP-hardness [DLY91] P2jpmtn; chainsjC j / J2jchains; p ij = 1jC j [LLLRK84] 1jpmtn; r j jw j C j / J2jr j ; p ij = 1jw j C j [L77],[LRKB77] 1jpmtnjw j T j / J2jp ij = 1jw j T j [DLY91] P2jchainsjC max / J2jno wait; chains; p ij = 1jC max [DLY91] P2jchainsjC j / J2jno wait; chains; p ij = 1jC j [LRKB77] 1jr j jC j / J2jno wa... |

32 |
Minimizing mean flow time with release time constraint
- Du, Leung, et al.
- 1990
(Show Context)
Citation Context ...ms listed in the second column are NP-hard or strongly NPhard as it has been established in the papers, references to which are listed in the first column. Table 4. Reductions proving the NP-hardness =-=[DLY90]-=- P2jpmtn; r j j\SigmaC j / J2jr j ; p ij = 1j\SigmaC j [DLW92] P2jpmtn; r j j\SigmaU j / J2jr j ; p ij = 1j\SigmaU j [BCS74],[LRKB77] P2jpmtnj\Sigmaw j C j / J2jp ij = 1j\Sigmaw j C j [K72] 1jpmtnj\Si... |

29 |
Scheduling Theory. Multi-Stage Systems
- Tanaev, Sotskov, et al.
- 1994
(Show Context)
Citation Context ...T j [?] J2jp ij = 1jw j U j [K96A],[?] J2jno wait; p ij = 1jC max [T85],[SL86],[K89],[?] J2jno wait; p ij = 1jw j C j [?] Table 3. Strongly NP-hard problems J2jchains; m j = 1; p ij = 1jC max [T85],[T=-=-=-SS94] J2jchains; p ij = 1jC j [?] J2jr j ; p ij = 1jw j C j [?] J2jp ij = 1jw j T j [?] J2jno wait; chains; p ij = 1jC max [?] J2jno wait; chains; p ij = 1jC j [?] J2jno wait; r j ; p ij = 1jC j [?] J... |

28 |
Rinnooy Kan. Preemptive scheduling of uniform machines subject to release dates
- Labetoulle, Lawler, et al.
- 1984
(Show Context)
Citation Context ...ax / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2kw j C j / J2jno wait; p ij = 1jw j C j Table 5. Reductions proving the strong NP-hardness [DLY91] P2jpmtn; chainsjC j / J2jchains; p ij = 1jC j [LLL=-=-=-RK84] 1jpmtn; r j jw j C j / J2jr j ; p ij = 1jw j C j [L77],[LRKB77] 1jpmtnjw j T j / J2jp ij = 1jw j T j [DLY91] P2jchainsjC max / J2jno wait; chains; p ij = 1jC max [DLY91] P2jchainsjC j / J2jno wa... |

27 |
Scheduling chainstructured tasks to minimize makespan and mean flow time
- Du, Leung, et al.
- 1991
(Show Context)
Citation Context ... j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC max / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2kw j C j / J2jno wait; p ij = 1jw j C j Table 5. Reductions proving the strong NP-hardness [DLY91] P2jpmtn; chainsjC j / J2jchains; p ij = 1jC j [LLLRK84] 1jpmtn; r j jw j C j / J2jr j ; p ij = 1jw j C j [L77],[LRKB77] 1jpmtnjw j T j / J2jp ij = 1jw j T j [DLY91] P2jchainsjC max / J2jno wait... |

25 |
Preemptive Scheduling with Release Times, Deadlines, and Due Dates
- Martel
- 1982
(Show Context)
Citation Context ...mal open problem. Theorem 3.1 proves the reduction P2jpmtn; r j jL max / J2jr j ; p ij = 1jL max , however this says nothing about the complexity of the latter problem because even Qjpmtn; r j jL max =-=[M8-=-2],[FG86] is solvable in polynomial time. Theorem 3.1 can be useful in identical-parallel-machines scheduling. For example, in the case of bounded number of jobs the reduction P jpmtn; prec; r j ; n ... |

20 |
Private communication
- Lenstra
- 2007
(Show Context)
Citation Context ...1jC max [?] J2jno wait; chains; p ij = 1jC j [?] J2jno wait; r j ; p ij = 1jC j [?] J2jno wait; r j ; p ij = 1jL max [?] J2jno wait; p ij = 1jw j T j [?] J3jp ij = 1jC max [LRK79] J3jp ij = 1jC j [L] J3jno wait; p ij = 1jC max [SL86] J3jno wait; p ij = 1jC j [SL86] Note that Kravchenko's algorithm [K96A] for J2jp ij = 1jU j uses as a subroutine an algorithm for J2jp ij = 1jL max which is equiva... |

19 |
Polynomial algorithms for resource-constrained and multiprocessor task scheduling problems
- Brucker, Kramer
- 1996
(Show Context)
Citation Context ...max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96A] J2jno wait; p ij = 1jC j [K96B] J jprec; r j ; n n; p ij = 1jw j T j [BK96] J jprec; r j ; n n; p ij = 1jw j U j [BK96] 4 TIMKOVSKY Table 2. NP-hard problems J2jr j ; p ij = 1jC j [?] J2jr j ; p ij = 1jU j [?] J2jp ij = 1jw j C j [?] J2jp ij = 1jT j [?] J2jp ij = 1j... |

16 |
Preemptive scheduling of uniform machines by ordinary network °ow techniques
- Federgruen, Groenevelt
- 1986
(Show Context)
Citation Context ...en problem. Theorem 3.1 proves the reduction P2jpmtn; r j jL max / J2jr j ; p ij = 1jL max , however this says nothing about the complexity of the latter problem because even Qjpmtn; r j jL max [M82],=-=[FG-=-86] is solvable in polynomial time. Theorem 3.1 can be useful in identical-parallel-machines scheduling. For example, in the case of bounded number of jobs the reduction P jpmtn; prec; r j ; n njw j... |

14 |
Rinnooy Kan. Recent developement in deterministic sequencing and scheduling
- Lawler, Lenstra, et al.
- 1982
(Show Context)
Citation Context ...2jno wait; p ij = 1jC max / P2jjC max J2jp ij = 1jC max / P2jpmtnjC max In what follow, we use the well-known three-eld classication jjsfor scheduling problems described in the surveys [GLLRK79], [LLR=-=K82-=-] and [LLRKS93], where ,sandsspecify the machine environment, job characteristics and the minimization criterion, respectively. The above two reductions allow to suggest that under the unit-processing... |

14 |
The complexity of shop-scheduling problems with two or three jobs
- SOTSKOV
- 1991
(Show Context)
Citation Context ...which is equivalent to J2jr j ; p ij = 1jC max by symmetry. Among job-shop problems with bounded number of jobs J2jn njw j T j [BKS95], J2jn njw j U j [BKS95], J jprec; r j ; n 2jw j T j [S91] and J jprec; r j ; n 2jw j U j [S91] are solvable in polynomial time. Therefore, their unit-time restricted versions are also so. Other complexity results on job-shop scheduling with arbitrary pro... |

13 |
Rinnooy Kan, A.: Computational complexity of discrete optimization
- Lenstra
- 1979
(Show Context)
Citation Context ...with the performance ratio 1 + (3m + h)=n [T86], where h is the number of operations in each job. Besides, C2jp ij 2 f1; 2gjC max , a restricted version of the strongly NP-hard J2jp ij 2 f1; 2gjC max =-=[LRK-=-79] is also so [T81]. Table 1. Problems solvable in polynomial time Cjp ij = 1jC max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96... |

11 | Preemptive scheduling of precedence-constrained jobs on parallel machines, deterministic and stochastic scheduling - Lawler - 1981 |

9 | A polynomial-time algorithm for the two-machine unit-time release-date job-shop schedule-length problem
- Timkovsky
- 1997
(Show Context)
Citation Context ...the strongly NP-hard J2jp ij 2 f1; 2gjC max [LRK79] is also so [T81]. Table 1. Problems solvable in polynomial time Cjp ij = 1jC max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96A] J2jno wait; p ij = 1jC j [K96B] J jprec; r j ; n n; p ij = 1jw j T j [BK96] J jprec; r j ; n n; p ij = 1jw j U j [BK96] 4 TIMKOVSKY Table 2.... |

8 |
Minimizing mean time with release time constraint
- Du, Leung, et al.
- 1990
(Show Context)
Citation Context ...ems listed in the second column are NP-hard or strongly NP-hard as it has been established in the papers, references to which are listed in thesrst column. Table 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [... |

6 |
Minimizing the Number of Late Jobs with Release Time Constraints
- Du, Leung, et al.
- 1992
(Show Context)
Citation Context ...gly NP-hard as it has been established in the papers, references to which are listed in thesrst column. Table 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC m... |

5 |
An ecient algorithm for a job shop problem
- Kubiak, Sethi, et al.
- 1995
(Show Context)
Citation Context ...ynomial-time algorithmsnding a nonpreemptive schedule to minimize the same criterion on two identical machines with arbitrary job processing times. As it was shown by Kubiak, Sethi and Sriskandarajah =-=[KSS95]-=- the same approach works out in developing a polynomial-time algorithm for the relaxed version of the former problem without the no-wait constraint employing a polynomial-time algorithm for the relaxe... |

5 |
A polynomial-time algorithm for total completion time minimization in two-machine job-shop with unit-time operations, EJOR 94
- Kubiak, Timkovsky
- 1996
(Show Context)
Citation Context ...jp ij 2 f1; 2gjC max [LRK79] is also so [T81]. Table 1. Problems solvable in polynomial time Cjp ij = 1jC max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96A] J2jno wait; p ij = 1jC j [K96B] J jprec; r j ; n n; p ij = 1jw j T j [BK96] J jprec; r j ; n n; p ij = 1jw j U j [BK96] 4 TIMKOVSKY Table 2. NP-hard problems J2jr j... |

5 |
On the complexity of scheduling an arbitrary system
- Timkovsky
- 1985
(Show Context)
Citation Context ...ds, then s i denotes the speed of the i th machine which has to process the ith operation in each job. The complexity status of unit-time cycle-shop problems except well-solvable Cjp ij = 1jC max [T=-=8-=-5] and C2jp ij = 1=s i jC max [T85] is unknown. A related result is a polynomialtime algorithm for Cjp ij = 1=s i jC maxsnding a schedule with the performance ratio 1 + (3m + h)=n [T86], where h is ... |

5 |
The complexity of unit-time job-shop scheduling
- Timkovsky
- 1993
(Show Context)
Citation Context ...chine. Evidently, the former problem is harder, and even with unit-processing-time operations it appears to have at least the same complexity as the latter problem with arbitrary job processing times =-=[T93-=-]. This paper makes an attempt to conrm this observation theoretically by describing for a wide class of minimization criteria a reduction from any m-identical-parallel-machines problem to an m-machin... |

4 |
On the complexity of two-machine jobshop scheduling with regular objective functions
- Brucker, Kravchenko, et al.
- 1997
(Show Context)
Citation Context ...ij = 1jU j uses as a subroutine an algorithm for J2jp ij = 1jL max which is equivalent to J2jr j ; p ij = 1jC max by symmetry. Among job-shop problems with bounded number of jobs J2jn njw j T j [BKS95], J2jn njw j U j [BKS95], J jprec; r j ; n 2jw j T j [S91] and J jprec; r j ; n 2jw j U j [S91] are solvable in polynomial time. Therefore, their unit-time restricted versions are also so. ... |

4 |
Scheduling independent tasks to reduce mean time
- Bruno, Jr, et al.
- 1974
(Show Context)
Citation Context ..., references to which are listed in thesrst column. Table 4. Reductions proving the NP-hardness [DLY90] P2jpmtn; r j jC j / J2jr j ; p ij = 1jC j [DLW92] P2jpmtn; r j jU j / J2jr j ; p ij = 1jU j [BCS74],[LRKB77] P2jpmtnjw j C j / J2jp ij = 1jw j C j [K72] 1jpmtnjw j U j / J2jp ij = 1jw j U j [DL90] 1jpmtnjT j / J2jp ij = 1jT j [LRKB77] P2kC max / J2jno wait; p ij = 1jC max [BCS74],[LRKB77] P2k... |

4 |
Minimizing the number of late jobs for the two-machine unit-time job-shop scheduling problem
- Kravchenko
- 1999
(Show Context)
Citation Context ...teristicsseld denotes asxed upper bound for the number n of jobs. The complexity status of J2jr j ; p ij = 1jL max is unclear. Pseudopolynomial-time algorithms are known only for J2jp ij = 1jw j U j [=-=K96A]-=- and J2jno wait; p ij = 1jC max [K89]. Hence, only these two problems are proved to be ordinarily NP-hard. The other NP-hard problems in Table 2 as well as the NP-hard J2jno wait; r j ; p ij = 1jC max... |

3 |
A polynomial algorithm for a two-machine no-wait jobshop scheduling problem
- Kravchenko
(Show Context)
Citation Context ...roblems solvable in polynomial time Cjp ij = 1jC max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96A] J2jno wait; p ij = 1jC j [K96B] J jprec; r j ; n n; p ij = 1jw j T j [BK96] J jprec; r j ; n n; p ij = 1jw j U j [BK96] 4 TIMKOVSKY Table 2. NP-hard problems J2jr j ; p ij = 1jC j [?] J2jr j ; p ij = 1jU j [?] J2jp ij =... |

3 |
A pseudopolynomial algorithm for a two-machine no-wait job shop problem, EJOR 43
- Kubiak
- 1989
(Show Context)
Citation Context ...lel-machines problem to an m-machine unit-time job-shop problem. It is interesting to note that in earlier works there have been described inverse reductions in the two-machine case. Probably, Kubiak =-=[K89]-=- was thesrst tosnd out such a Key words and phrases. Scheduling, polynomial-time reduction, NP-hardness, identical parallel machines, job shop, unit-time operations. y This work was partially supporte... |

3 |
Jr.: Optimal preemptive scheduling on two-processor systems
- Muntz, Coffman
- 1969
(Show Context)
Citation Context ... or ordinarily NP-hard problems to strongly NP-hard problems. It proves that some unit-time job-shop problems are harder than their identical-parallel-machines counterparts, unless P=NP. For example, =-=[M-=-C69],[MC70] P2jpmtn; precjC max / J2jprec; p ij = 1jC max [T85],[TSS94] [M59] P3jpmtnjC max / J3jp ij = 1jC max [LRK79] [M59] P3jpmtnjC j / J3jp ij = 1jC j [L] [CMM67] P3kC j / J3jno wait; p ij = 1jC ... |

3 |
Coffman Jr, Preemptive scheduling of real time tasks on multiprocessor systems
- Muntz, G
- 1970
(Show Context)
Citation Context ...inarily NP-hard problems to strongly NP-hard problems. It proves that some unit-time job-shop problems are harder than their identical-parallel-machines counterparts, unless P=NP. For example, [MC69],=-=[M-=-C70] P2jpmtn; precjC max / J2jprec; p ij = 1jC max [T85],[TSS94] [M59] P3jpmtnjC max / J3jp ij = 1jC max [LRK79] [M59] P3jpmtnjC j / J3jp ij = 1jC j [L] [CMM67] P3kC j / J3jno wait; p ij = 1jC j [SL86... |

3 |
Some no-wait jobs scheduling problems: complexity results, EJOR 24
- Sriskandarajah, Ladet
- 1986
(Show Context)
Citation Context ...IMKOVSKY Table 2. NP-hard problems J2jr j ; p ij = 1jC j [?] J2jr j ; p ij = 1jU j [?] J2jp ij = 1jw j C j [?] J2jp ij = 1jT j [?] J2jp ij = 1jw j U j [K96A],[?] J2jno wait; p ij = 1jC max [T85],[SL86],[K89],[?] J2jno wait; p ij = 1jw j C j [?] Table 3. Strongly NP-hard problems J2jchains; m j = 1; p ij = 1jC max [T85],[TSS94] J2jchains; p ij = 1jC j [?] J2jr j ; p ij = 1jw j C j [?] J2jp ij = 1... |

2 |
Computational complexity and approximation of the cycle shop scheduling problem
- Timkovsky
- 1981
(Show Context)
Citation Context ...e ratio 1 + (3m + h)=n [T86], where h is the number of operations in each job. Besides, C2jp ij 2 f1; 2gjC max , a restricted version of the strongly NP-hard J2jp ij 2 f1; 2gjC max [LRK79] is also so =-=[T-=-81]. Table 1. Problems solvable in polynomial time Cjp ij = 1jC max [T85] C2jp ij = 1=s i jC max [T85] J2jr j ; p ij = 1jC max [T93],[T97] J2jp ij = 1jC j [KT96] J2jp ij = 1jU j [K96A] J2jno wait; p ... |

1 | An approximation for the cycle shop scheduling problem, Ekonomika i Matematicheskije Metodi 22 - Timkovsky - 1986 |