An Offline and Online Algorithm for All Minimal k|U| Parameter Subsets of a Soft Set Based on Integer Partition

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.