### Abstract

In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by Δ_{i} the number of non-zero coefficients in the ith constraints. Furthermore, we assume that Δ_{1}≥ Δ_{2}≥ ⋯ ≥ Δ_{m}. For this problem, Koufogiannakis and Young proposed a polynomial-time Δ_{1}-approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with { 0 , 1 } -variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is max { Δ_{2}, min { Δ_{1}, 1 + Π} } , where Π is the maximum size of a connected component of the input submodular function.

Original language | English |
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Pages (from-to) | 2957-2971 |

Number of pages | 15 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

**A Note on Submodular Function Minimization with Covering Type Linear Constraints.** / Kamiyama, Naoyuki.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 80, no. 10, pp. 2957-2971. https://doi.org/10.1007/s00453-017-0363-8

}

TY - JOUR

T1 - A Note on Submodular Function Minimization with Covering Type Linear Constraints

AU - Kamiyama, Naoyuki

PY - 2018/10/1

Y1 - 2018/10/1

N2 - In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by Δi the number of non-zero coefficients in the ith constraints. Furthermore, we assume that Δ1≥ Δ2≥ ⋯ ≥ Δm. For this problem, Koufogiannakis and Young proposed a polynomial-time Δ1-approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with { 0 , 1 } -variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is max { Δ2, min { Δ1, 1 + Π} } , where Π is the maximum size of a connected component of the input submodular function.

AB - In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by Δi the number of non-zero coefficients in the ith constraints. Furthermore, we assume that Δ1≥ Δ2≥ ⋯ ≥ Δm. For this problem, Koufogiannakis and Young proposed a polynomial-time Δ1-approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with { 0 , 1 } -variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is max { Δ2, min { Δ1, 1 + Π} } , where Π is the maximum size of a connected component of the input submodular function.

UR - http://www.scopus.com/inward/record.url?scp=85028537386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028537386&partnerID=8YFLogxK

U2 - 10.1007/s00453-017-0363-8

DO - 10.1007/s00453-017-0363-8

M3 - Article

AN - SCOPUS:85028537386

VL - 80

SP - 2957

EP - 2971

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 10

ER -