# Heuristic Approaches to Attribute Reduction for Generalized Decision Preservation
## 1 Introduction  
Originating from the mathematician [PERSON] in the early 1980s, rough set theory (RST) [1] has been regarded as an effective tool for the processing of inconsistent or uncertain information. It has been extensively utilized in research fields such as uncertainty reasoning [2; 3], knowledge representation [4], feature selection, and machine learning [5; 6; 7]. Attribute reduction [1; 8; 9; 10] plays a crucial role in RST. The main purpose of attribute reduction is to find a minimal attribute subset which has the same classification or discernibility properties as the original information system. Different attribute reductions can leave different classification or discernibility properties of information systems unchanged. For example, positive region preservation reduction [11] can leave the positive region of a target decision unchanged. Generalized decision preservation reduction [12; 13] can leave the generalized decision of each object in the universe unchanged. Mutual information preservation reduction [14] can leave mutual information, with respect to the decision attributes, unchanged. In the last twenty years, many methods for attribute reduction have been studied, such as discernibility matrix-based attribute reduction methods [15; 16; 17; 18; 19; 20], heuristic attribute reduction methods [21; 22; 23; 10], metaheuristic attribute reduction methods [24; 25; 26; 27; 28; 29; 30], and so on. In attribute reduction, the discernibility matrix is an important technique for obtaining all reducts from data sets. [PERSON] et al. [8] first proposed a discernibility matrix which can obtain all reducts from an information system. To extend the classical discernibility matrix, many methods of discernibility matrix-based attribute reduction have been studied. Based on the similarity relation, [PERSON] [13] proposed discernibility matrices in incomplete information systems. By using maximal consistentblocks in incomplete information systems, [PERSON] et al. [31] proposed a more efficient computational method for attribute reduction. [PERSON] et al. [32] introduced an attribute reduction method which can leave maximal consistent blocks unchanged in an interval-valued information system. To obtain the reducts, with respect to one decision class instead of all decision classes, [PERSON] et al. [33] constructed discernibility matrices regarding \(l\)th lower approximation reduction, \(l\)th decision class reduction, and \(\beta\)-reduction of the \(l\)th decision class. [PERSON] et al. [12] proposed a theoretical framework for discernibility matrix-based attribute reduction. Based on this framework, the generalized discernibility matrix was constructed. As mentioned above, all reducts can be found by using discernibility matrices. However, finding all reducts from a decision system based on a discernibility matrix has been proved to be an NP-hard problem. Therefore, the aforementioned methods based on discernibility matrices are inefficient and difficult to apply in large-scale data sets. To increase the efficiency of attribute reduction, many methods of heuristic attribute reduction and metaheuristic attribute reduction have been researched extensively. [PERSON] et al. [11] first introduced a heuristic method for positive region reduction. Using the mutual information, [PERSON] et al. [14] proposed a bottom-up reduction algorithm. [PERSON] et al. [34] closely focused on increasing the efficiencies of heuristic reduction algorithms, and adopted a positive approximation strategy to accelerate heuristic reduction algorithms. [PERSON] et al. [35] used a variant form of conditional entropy to design an attribute reduction algorithm for an interval-valued decision system. Many attribute reduction algorithms based on metaheuristic methods [24; 25; 26; 27; 28; 29; 30; 36; 37] have been developed recently. [PERSON] et al. [26] used a genetic algorithm to obtain a global minimal reduct in a decision-theoretic rough set model. By using ant colony optimization, [PERSON] et al.