### Abstract

A number of combinatorial optimization problems in machine learning can be described as the problem of minimizing a submodular function. It is known that the unconstrained submodular minimization problem can be solved in strongly polynomial time. However, additional constraints make the problem intractable in many settings. In this paper, we discuss the submodular minimization under a size constraint, which is NP-hard, and generalizes the densest subgraph problem and the uniform graph partitioning problem. Because of NP-hardness, it is difficult to compute an optimal solution even for a prescribed size constraint. In our approach, we do not give approximation algorithms. Instead, the proposed algorithm computes optimal solutions for some of possible size constraints in polynomial time. Our algorithm utilizes the basic polyhedral theory associated with submodular functions. Additionally, we evaluate the performance of the proposed algorithm through computational experiments.

Original language | English |
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Title of host publication | Proceedings of the 28th International Conference on Machine Learning, ICML 2011 |

Pages | 977-984 |

Number of pages | 8 |

Publication status | Published - Oct 7 2011 |

Externally published | Yes |

Event | 28th International Conference on Machine Learning, ICML 2011 - Bellevue, WA, United States Duration: Jun 28 2011 → Jul 2 2011 |

### Publication series

Name | Proceedings of the 28th International Conference on Machine Learning, ICML 2011 |
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### Conference

Conference | 28th International Conference on Machine Learning, ICML 2011 |
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Country | United States |

City | Bellevue, WA |

Period | 6/28/11 → 7/2/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Human-Computer Interaction
- Education

### Cite this

*Proceedings of the 28th International Conference on Machine Learning, ICML 2011*(pp. 977-984). (Proceedings of the 28th International Conference on Machine Learning, ICML 2011).

**Size-constrained submodular minimization through minimum norm base.** / Nagano, Kiyohito; Kawahara, Yoshinobu; Aihara, Kazuyuki.

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

*Proceedings of the 28th International Conference on Machine Learning, ICML 2011.*Proceedings of the 28th International Conference on Machine Learning, ICML 2011, pp. 977-984, 28th International Conference on Machine Learning, ICML 2011, Bellevue, WA, United States, 6/28/11.

}

TY - GEN

T1 - Size-constrained submodular minimization through minimum norm base

AU - Nagano, Kiyohito

AU - Kawahara, Yoshinobu

AU - Aihara, Kazuyuki

PY - 2011/10/7

Y1 - 2011/10/7

N2 - A number of combinatorial optimization problems in machine learning can be described as the problem of minimizing a submodular function. It is known that the unconstrained submodular minimization problem can be solved in strongly polynomial time. However, additional constraints make the problem intractable in many settings. In this paper, we discuss the submodular minimization under a size constraint, which is NP-hard, and generalizes the densest subgraph problem and the uniform graph partitioning problem. Because of NP-hardness, it is difficult to compute an optimal solution even for a prescribed size constraint. In our approach, we do not give approximation algorithms. Instead, the proposed algorithm computes optimal solutions for some of possible size constraints in polynomial time. Our algorithm utilizes the basic polyhedral theory associated with submodular functions. Additionally, we evaluate the performance of the proposed algorithm through computational experiments.

AB - A number of combinatorial optimization problems in machine learning can be described as the problem of minimizing a submodular function. It is known that the unconstrained submodular minimization problem can be solved in strongly polynomial time. However, additional constraints make the problem intractable in many settings. In this paper, we discuss the submodular minimization under a size constraint, which is NP-hard, and generalizes the densest subgraph problem and the uniform graph partitioning problem. Because of NP-hardness, it is difficult to compute an optimal solution even for a prescribed size constraint. In our approach, we do not give approximation algorithms. Instead, the proposed algorithm computes optimal solutions for some of possible size constraints in polynomial time. Our algorithm utilizes the basic polyhedral theory associated with submodular functions. Additionally, we evaluate the performance of the proposed algorithm through computational experiments.

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M3 - Conference contribution

AN - SCOPUS:80053442724

SN - 9781450306195

T3 - Proceedings of the 28th International Conference on Machine Learning, ICML 2011

SP - 977

EP - 984

BT - Proceedings of the 28th International Conference on Machine Learning, ICML 2011

ER -