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A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

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Abstract

In this paper, we consider the k-prize-collecting minimum vertex cover problem with submodular penalties, which generalizes the well-known minimum vertex cover problem, minimum partial vertex cover problem and minimum vertex cover problem with submodular penalties. We are given a cost graph G = (V, E; c) and an integer k. This problem determines a vertex set S ⊆ V such that S covers at least k edges. The objective is to minimize the total cost of the vertices in S plus the penalty of the uncovered edge set, where the penalty is determined by a submodular function. We design a two-phase combinatorial algorithm based on the guessing technique and the primal-dual framework to address the problem. When the submodular penalty cost function is normalized and nondecreasing, the proposed algorithm has an approximation factor of 3. When the submodular penalty cost function is linear, the approximation factor of the proposed algorithm is reduced to 2, which is the best factor if the unique game conjecture holds.

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Acknowledgements

The work was supported in part by the National Natural Science Foundation of China (Grant No. 12071417).

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Yunnan University, Kunming, 650500, China

    Xiaofei Liu

  2. School of Mathematics and Statistics, Yunnan University, Kunming, 650500, China

    Weidong Li

  3. Dianchi College of Yunnan University, Kunming, 650228, China

    Jinhua Yang

Authors
  1. Xiaofei Liu
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  2. Weidong Li
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  3. Jinhua Yang
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Corresponding author

Correspondence to Weidong Li.

Additional information

Xiaofei Liu received PhD degree in Mathematics from Yunnan University, China in 2018. He was a postdoctoral researcher at Peking University, China. He is currently a lecturer with Yunnan University, China. His research interests include theoretical computer science and discrete optimization.

Weidong Li received the PhD degree in Department of Mathematics, from Yunnan University, China in 2010. He is currently a professor at Yunnan University, China. His research interests include discrete optimization and algorithmic game theory.

Jinhua Yang is currently an associate professor at Dianchi College of Yunnan University, China. His research interests include image processing, deep learning and machine learning.

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Liu, X., Li, W. & Yang, J. A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties. Front. Comput. Sci. 17, 173404 (2023). https://doi.org/10.1007/s11704-022-1665-9

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  • DOI: https://doi.org/10.1007/s11704-022-1665-9

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