### Abstract

In this note, we investigate the time complexity of non-preemptive shop scheduling problems with two jobs. First we study mixed shop scheduling where one job has a fixed order of operations and the operations of the other job may be executed in arbitrary order. This problem is shown to be binary NP-complete with respect to all traditional optimality criteria even if distinct operations of the same job require different machines. Moreover, we devise a pseudo-polynomial time algorithm which solves the problem with respect to all non-decreasing objective functions. Finally, when the job with fixed order of operations may visit a machine more than once, the problem becomes unary NP-complete. Then we discuss shop scheduling with two jobs having chain-like routings. It is shown that the problem is unary NP-complete with respect to all traditional optimality criteria even if one of the jobs has fixed order of operations and the jobs cannot visit a machine twice.

Original language | English |
---|---|

Pages (from-to) | 37-49 |

Number of pages | 13 |

Journal | Computing |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 |

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### Keywords

- Computational complexity
- Mixed shop scheduling
- Pseudo-polynomial time algorithm

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

**On the complexity of non-preemptive shop scheduling with two jobs.** / Kis, T.

Research output: Contribution to journal › Article

*Computing*, vol. 69, no. 1, pp. 37-49. https://doi.org/10.1007/s00607-002-1455-z

}

TY - JOUR

T1 - On the complexity of non-preemptive shop scheduling with two jobs

AU - Kis, T.

PY - 2002

Y1 - 2002

N2 - In this note, we investigate the time complexity of non-preemptive shop scheduling problems with two jobs. First we study mixed shop scheduling where one job has a fixed order of operations and the operations of the other job may be executed in arbitrary order. This problem is shown to be binary NP-complete with respect to all traditional optimality criteria even if distinct operations of the same job require different machines. Moreover, we devise a pseudo-polynomial time algorithm which solves the problem with respect to all non-decreasing objective functions. Finally, when the job with fixed order of operations may visit a machine more than once, the problem becomes unary NP-complete. Then we discuss shop scheduling with two jobs having chain-like routings. It is shown that the problem is unary NP-complete with respect to all traditional optimality criteria even if one of the jobs has fixed order of operations and the jobs cannot visit a machine twice.

AB - In this note, we investigate the time complexity of non-preemptive shop scheduling problems with two jobs. First we study mixed shop scheduling where one job has a fixed order of operations and the operations of the other job may be executed in arbitrary order. This problem is shown to be binary NP-complete with respect to all traditional optimality criteria even if distinct operations of the same job require different machines. Moreover, we devise a pseudo-polynomial time algorithm which solves the problem with respect to all non-decreasing objective functions. Finally, when the job with fixed order of operations may visit a machine more than once, the problem becomes unary NP-complete. Then we discuss shop scheduling with two jobs having chain-like routings. It is shown that the problem is unary NP-complete with respect to all traditional optimality criteria even if one of the jobs has fixed order of operations and the jobs cannot visit a machine twice.

KW - Computational complexity

KW - Mixed shop scheduling

KW - Pseudo-polynomial time algorithm

UR - http://www.scopus.com/inward/record.url?scp=0036392349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036392349&partnerID=8YFLogxK

U2 - 10.1007/s00607-002-1455-z

DO - 10.1007/s00607-002-1455-z

M3 - Article

AN - SCOPUS:0036392349

VL - 69

SP - 37

EP - 49

JO - Computing (Vienna/New York)

JF - Computing (Vienna/New York)

SN - 0010-485X

IS - 1

ER -