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A polynomial time algorithm for finding a minimum 4-partition of a submodular function

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In this paper, we study the minimum k-partition problem of submodular functions, i.e., given a finite set V and a submodular function \(f:2^V\rightarrow \mathbb {R}\), computing a k-partition \( \{ V_1, \ldots , V_k \}\) of V with minimum \(\sum _{i=1}^k f(V_i)\). The problem is a natural generalization of the minimum k-cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k, and solvable in polynomial time for fixed \(k \le 3\). In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem.

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Acknowledgements

Chao would like to thank Chandra Chekuri and Karthekeyan Chandrasekaran for early discussions.

Funding

This work was partially supported by JST ERATO Grant Number JPMJER2301 and Japan Society for the Promotion of Science (JSPS) KAK420 ENHI Grant Numbers JP19K22841, JP20H00609, and JP20H05967.

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Authors and Affiliations

  1. Toshiba Digital Solutions Corporation, Kawasaki, Japan

    Tsuyoshi Hirayama

  2. University of Electronic Science and Technology of China, Chengdu, China

    Yuhao Liu, Ke Shi & Chao Xu

  3. Kyoto University, Kyoto, Japan

    Kazuhisa Makino

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  1. Tsuyoshi Hirayama
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Correspondence to Chao Xu.

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Authors are ordered alphabetically. This work was partially supported by JST ERATO Grant Number JPMJER2301 and JSPS KAKENHI Grant Numbers JP19K22841, JP20H00609, and JP20H05967. An earlier version of this paper appeared in SODA 2023 [17]. This version introduces have a more general algorithm for \((1,\ell )\)-size 3-partition, and a graph example at the end.

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Hirayama, T., Liu, Y., Makino, K. et al. A polynomial time algorithm for finding a minimum 4-partition of a submodular function. Math. Program. 207, 717–732 (2024). https://doi.org/10.1007/s10107-023-02029-0

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