A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets

Efficient attribute reduction in large, incomplete decision systems is a challenging problem; existing approaches have time complexities no less than O(∣C∣2∣U∣2). This paper derives some important properties of incomplete information systems, then constructs a positive region-based algorithm to solve the attribute reduction problem with a time complexity no more than O(∣C∣2∣U∣log∣U∣). Furthermore, our approach does not change the size of the original incomplete system. Numerical experiments show that the proposed approach is indeed efficient, and therefore of practical value to many real-world problems. The proposed algorithm can be applied to both consistent and inconsistent incomplete decision systems.