TY - JOUR

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

AU - Kamiyama, Naoyuki

N1 - Funding Information:
Acknowledgements The author would like to thank Naonori Kakimura and the anonymous referees for helpful comments. This research was supported by JST PRESTO Grant Number JPMJPR14E1, Japan.
Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

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.

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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 -