Abstract
The hitting set problem is a generalization of the vertex cover problem to hypergraphs. Xu et al. (Theor Comput Sci 630:117–125, 2016) presented a primal-dual algorithm for the submodular vertex cover problem with linear/submodular penalties. Motivated by their work, we study the submodular hitting set problem with linear penalties (SHSLP). The goal of the SHSLP is to select a vertex subset in the hypergraph to cover some hyperedges and penalize the uncovered ones such that the total cost of covering and penalty is minimized. Based on the primal-dual scheme, we obtain a k-approximation algorithm for the SHSLP, where k is the maximum number of vertices in all hyperedges.
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Acknowledgements
The authors would like to thank the referees for giving this paper a careful reading and many valuable comments and useful suggestions. The paper was written while Professor Dingzhu Du visited Hebei Normal University and the authors would like to give their sincere thanks to him for his guidance and valuable ideas. This work is supported by the NSF of China (No. 11971146), the NSF of Hebei Province (Nos. A2019205089, A2019205092), Hebei Province Foundation for Returnees (CL201714) and Overseas Expertise Introduction Program of Hebei Auspices (25305008).
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Du, S., Gao, S., Hou, B. et al. An approximation algorithm for submodular hitting set problem with linear penalties. J Comb Optim 40, 1065–1074 (2020). https://doi.org/10.1007/s10878-020-00653-6
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DOI: https://doi.org/10.1007/s10878-020-00653-6