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A Note on Submodular Function Minimization with Covering Type Linear Constraints

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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 \(\varDelta _i\) the number of non-zero coefficients in the ith constraints. Furthermore, we assume that \(\varDelta _1 \ge \varDelta _2 \ge \cdots \ge \varDelta _m\). For this problem, Koufogiannakis and Young proposed a polynomial-time \(\varDelta _1\)-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 \(\max \{\varDelta _2, \min \{\varDelta _1, 1 + \varPi \}\}\), where \(\varPi \) is the maximum size of a connected component of the input submodular function.

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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.

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  1. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Naoyuki Kamiyama

  2. JST, PRESTO, Kawaguchi, Japan

    Naoyuki Kamiyama

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  1. Naoyuki Kamiyama
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Correspondence to Naoyuki Kamiyama.

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Kamiyama, N. A Note on Submodular Function Minimization with Covering Type Linear Constraints. Algorithmica 80, 2957–2971 (2018). https://doi.org/10.1007/s00453-017-0363-8

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