Abstract
As an important technique for granular computing, rough sets deal with vagueness and granularity in information systems. Rough sets are usually used in attribute reduction, however, the corresponding algorithms are often greedy ones. Matroids generalize the linear independence in vector spaces and provide well-established platforms for greedy algorithms. In this paper, we apply contraction to a matroidal structure of rough sets. Firstly, for an equivalence relation on a universe, a matroid is established through the lower approximation operator. Secondly, three characteristics of the dual of the matroid, which are useful for applying a new operation to the dual matroid, are investigated. Finally, the operation named contraction is applied to the dual matroid. We study some relationships between the contractions of the dual matroid to two subsets, which are the complement of a single point set and the complement of the equivalence class of this point. Moreover, these relationships are extended to general cases. In a word, these results show an interesting view to investigate the combination between rough sets and matroids.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chen, Y., Miao, D., Wang, R., Wu, K.: A rough set approach to feature selection based on power set tree. Knowledge-Based Systems 24, 275–281 (2011)
Dai, J., Wang, W., Xu, Q., Tian, H.: Uncertainty measurement for interval-valued decision systems based on extended conditional entropy. Knowledge-Based Systems 27, 443–450 (2012)
Dougherty, R., Freiling, C., Zeger, K.: Networks, matroids, and non-shannon information inequalities. IEEE Transactions on Information Theory 53, 1949–1969 (2007)
Du, Y., Hu, Q., Zhu, P., Ma, P.: Rule learning for classification based on neighborhood covering reduction. Information Sciences 181, 5457–5467 (2011)
Edmonds, J.: Matroids and the greedy algorithm. Mathematical Programming 1, 127–136 (1971)
Hu, Q., Yu, D., Liu, J., Wu, C.: Neighborhood rough set based heterogeneous feature subset selection. Information Sciences 178, 3577–3594 (2008)
Jia, X., Liao, W., Tang, Z., Shang, L.: Minimum cost attribute reduction in decision-theoretic rough set models. Information Sciences 219, 151–167 (2013)
Kryszkiewicz, M.: Rough set approach to incomplete information systems. Information Sciences 112, 39–49 (1998)
Lai, H.: Matroid theory. Higher Education Press, Beijing (2001)
Lawler, E.: Combinatorial optimization: networks and matroids. Dover Publications (2001)
Leung, Y., Fung, T., Mi, J., Wu, W.: A rough set approach to the discovery of classification rules in spatial data. International Journal of Geographical Information Science 21, 1033–1058 (2007)
Li, Y.: Some researches on fuzzy matroids. PhD thesis, Shaanxi Normal University (2007)
Mao, H.: The relation between matroid and concept lattice. Advances in Mathematics 35, 361–365 (2006)
Matus, F.: Abstract functional dependency structures. Theoretical Computer Science 81, 117–126 (1991)
Min, F., Zhu, W.: Attribute reduction of data with error ranges and test costs. Information Sciences 211, 48–67 (2012)
Pawlak, Z.: Rough sets. ICS PAS Reports 431 (1981)
Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)
Qin, K., Yang, J., Pei, Z.: Generalized rough sets based on reflexive and transitive relations. Information Sciences 178, 4138–4141 (2008)
Rouayheb, S.Y.E., Sprintson, A., Georghiades, C.N.: On the index coding problem and its relation to network coding and matroid theory. IEEE Transactions on Information Theory 56, 3187–3195 (2010)
Wang, S., Zhu, Q., Zhu, W., Min, F.: Matroidal structure of rough sets and its characterization to attribute reduction. Knowledge-Based Systems 36, 155–161 (2012)
Wang, S., Zhu, Q., Zhu, W., Min, F.: Quantitative analysis for covering-based rough sets using the upper approximation number. Information Sciences 220, 483–491 (2013)
Yao, Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111, 239–259 (1998)
Yao, Y.: Constructive and algebraic methods of theory of rough sets. Information Sciences 109, 21–47 (1998)
Zhong, N.: Rough sets in knowledge discovery and data mining. Journal of Japan Society for Fuzzy Theory and Systems 13, 581–591 (2001)
Zhu, W., Wang, F.: Reduction and axiomization of covering generalized rough sets. Information Sciences 152, 217–230 (2003)
Zhu, W., Wang, S.: Matroidal approaches to generalized rough sets based on relations. International Journal of Machine Learning and Cybernetics 2, 273–279 (2011)
Zhu, W., Wang, S.: Rough matroid based on relations. Information Sciences 232, 241–252 (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wang, J., Zhu, W. (2013). Contraction to Matroidal Structure of Rough Sets. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds) Rough Sets and Knowledge Technology. RSKT 2013. Lecture Notes in Computer Science(), vol 8171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41299-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-41299-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41298-1
Online ISBN: 978-3-642-41299-8
eBook Packages: Computer ScienceComputer Science (R0)