### Abstract

We consider a single machine scheduling problem to minimize total flow time under precedence constraints, which is NP-hard. Matsumoto et al. proposed an exact algorithm that consists of two phases: first construct a Multi-valued Decision Diagram (MDD) to represent feasible permutations of jobs, and then find the shortest path in the MDD which corresponds to the optimal solution. Although their algorithm performs significantly better than standard IP solvers for problems with dense constraints, the performance rapidly diminishes when the number of constraints decreases, which is due to the exponential growth of MDDs. In this paper, we introduce an equivalence relation among feasible permutations and show that it suffices to construct an MDD that maintains only one representative for each equivalence class. Experimental results show that our algorithm outperforms Matsumoto et al.’s algorithm for problems with sparse constraints, while keeping good performance for dense constraints. Moreover, we show that Matsumoto et al.’s algorithm can be extended for solving a more general problem of minimizing weighted total flow time.

Original language | English |
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Title of host publication | Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings |

Editors | Charles J. Colbourn, Roberto Grossi, Nadia Pisanti |

Publisher | Springer Verlag |

Pages | 365-377 |

Number of pages | 13 |

ISBN (Print) | 9783030250041 |

DOIs | |

Publication status | Published - Jan 1 2019 |

Event | 30th International Workshop on Combinatorial Algorithms, IWOCA 2019 - Pisa, Italy Duration: Jul 23 2019 → Jul 25 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11638 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 30th International Workshop on Combinatorial Algorithms, IWOCA 2019 |
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Country | Italy |

City | Pisa |

Period | 7/23/19 → 7/25/19 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings*(pp. 365-377). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11638 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-25005-8_30

**Succinct representation of linear extensions via MDDs and its application to scheduling under precedence constraints.** / Miyake, Fumito; Takimoto, Eiji; hatano, kohei.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11638 LNCS, Springer Verlag, pp. 365-377, 30th International Workshop on Combinatorial Algorithms, IWOCA 2019, Pisa, Italy, 7/23/19. https://doi.org/10.1007/978-3-030-25005-8_30

}

TY - GEN

T1 - Succinct representation of linear extensions via MDDs and its application to scheduling under precedence constraints

AU - Miyake, Fumito

AU - Takimoto, Eiji

AU - hatano, kohei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider a single machine scheduling problem to minimize total flow time under precedence constraints, which is NP-hard. Matsumoto et al. proposed an exact algorithm that consists of two phases: first construct a Multi-valued Decision Diagram (MDD) to represent feasible permutations of jobs, and then find the shortest path in the MDD which corresponds to the optimal solution. Although their algorithm performs significantly better than standard IP solvers for problems with dense constraints, the performance rapidly diminishes when the number of constraints decreases, which is due to the exponential growth of MDDs. In this paper, we introduce an equivalence relation among feasible permutations and show that it suffices to construct an MDD that maintains only one representative for each equivalence class. Experimental results show that our algorithm outperforms Matsumoto et al.’s algorithm for problems with sparse constraints, while keeping good performance for dense constraints. Moreover, we show that Matsumoto et al.’s algorithm can be extended for solving a more general problem of minimizing weighted total flow time.

AB - We consider a single machine scheduling problem to minimize total flow time under precedence constraints, which is NP-hard. Matsumoto et al. proposed an exact algorithm that consists of two phases: first construct a Multi-valued Decision Diagram (MDD) to represent feasible permutations of jobs, and then find the shortest path in the MDD which corresponds to the optimal solution. Although their algorithm performs significantly better than standard IP solvers for problems with dense constraints, the performance rapidly diminishes when the number of constraints decreases, which is due to the exponential growth of MDDs. In this paper, we introduce an equivalence relation among feasible permutations and show that it suffices to construct an MDD that maintains only one representative for each equivalence class. Experimental results show that our algorithm outperforms Matsumoto et al.’s algorithm for problems with sparse constraints, while keeping good performance for dense constraints. Moreover, we show that Matsumoto et al.’s algorithm can be extended for solving a more general problem of minimizing weighted total flow time.

UR - http://www.scopus.com/inward/record.url?scp=85069725088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069725088&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-25005-8_30

DO - 10.1007/978-3-030-25005-8_30

M3 - Conference contribution

SN - 9783030250041

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 365

EP - 377

BT - Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings

A2 - Colbourn, Charles J.

A2 - Grossi, Roberto

A2 - Pisanti, Nadia

PB - Springer Verlag

ER -