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Conditions for coverings to induce matroids

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Abstract

Coverings are a useful form of data structure, and covering-based rough sets provide an effective tool to cope with this type of data. However, many important problems such as covering reduction in covering-based rough sets are NP-hard, so that most algorithms to solve them are greedy ones. Matroids, as a generalization of the linear independence in vector spaces, provide well-established platforms for greedy algorithms. Therefore, it is necessary to integrate covering-based rough sets and matroids. In this paper, we present conditions for coverings to induce matroids. Firstly, some conditions under which the minimal set of a covering satisfies the circuit axiom of matroids are presented through three sides, which are coverings, matroids and neighborhoods, then a matroid is induced by the covering. Secondly, two conditions under which two different coverings can induce the same matroid are studied. Finally, two sufficient and necessary conditions for a neighborhood covering to induce an Eulerian matroid are investigated, where the neighborhood covering is a family of all neighborhoods. In a word, these results show an interesting view to investigate the combination between covering-based rough sets and matroids.

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Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant Nos. 61170128, 61379049 and 61379089, the Natural Science Foundation of Fujian Province, China, under Grant No. 2012J01294, and the Science and Technology Key Project of Fujian Province, China, under Grant No. 2012H0043.

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Authors and Affiliations

  1. Lab of Granular Computing, Minnan Normal University, Zhangzhou, 363000, China

    Jingqian Wang & William Zhu

  2. Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China

    Fei-Yue Wang

  3. College of Information Science, Beijing Language and Culture University, Beijing, 100871, China

    Guilong Liu

Authors
  1. Jingqian Wang
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  2. William Zhu
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  3. Fei-Yue Wang
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  4. Guilong Liu
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Correspondence to William Zhu.

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Wang, J., Zhu, W., Wang, FY. et al. Conditions for coverings to induce matroids. Int. J. Mach. Learn. & Cyber. 5, 947–954 (2014). https://doi.org/10.1007/s13042-014-0236-2

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  • DOI: https://doi.org/10.1007/s13042-014-0236-2

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