Abstract
This article investigates k|U| parameter subsets of a soft set matrix whose column sums are integral multiples of |U| (i.e., the number of objects in the soft set domain U). This kind of parameter subset represents an important data structure. Particularly, as a necessary condition, it has been shown to be useful in the parameter reduction problems of soft sets. This article focuses on the minimal k|U| parameter subsets, whose any proper subset cannot be a k|U| parameter subset. An offline and online algorithm for minimal k|U| parameter subsets is proposed. Its basic function is based on integer partition in an offline way. When soft set data come online, the algorithm only needs to filter the factorization results according to the related constraints within the input soft set. We also bring in combinatorial formulas for computing the number of k|U| parameter subsets and the approximate number of minimal k|U| parameter subsets. As an application of k|U| parameter subsets, the method of integer partition is also extended for normal parameter reduction problems of soft sets. The experimental results show that the proposed method does result in better performance.
Highlights
In 1999, Molodtsov introduced the theory of the soft set [1], which is a novel mathematical tool for dealing with uncertainties and vagueness
The k|U | parameter subsets of soft sets, whose sum of column values is an integral multiple of the number of rows, is a kind of data structure of the soft set matrix and plays an important role in solving the normal parameter reduction problems
In this article we focus on the minimal k|U | parameter subsets of soft sets
Summary
In 1999, Molodtsov introduced the theory of the soft set [1], which is a novel mathematical tool for dealing with uncertainties and vagueness. The k|U | parameter subsets of soft sets, whose sum of column values is an integral multiple of the number of rows (i.e, number of objects), is a kind of data structure of the soft set matrix and plays an important role in solving the normal parameter reduction problems. In [40], a hierarchical algorithm for computing all minimal k|U | parameter subsets is studied, and the normal parameter reduction problem of the soft set can be solved by testing the disjoint combinations of these minimal k|U | parameter subsets. We need to develop a much more efficient method for computing all minimal k|U | parameter subsets This will be beneficial for the knowledge mining and parameter reduction problems of soft sets. We will reach a conclusion of this article, indicate the novelty and potential weakness of our methods, and offer our outlook for potential future work
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