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

This article investigates k vertical bar U vertical bar parameter subsets of a soft set matrix whose column sums are integral multiples of vertical bar U vertical bar (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 vertical bar U vertical bar parameter subsets, whose any proper subset cannot be a k vertical bar U vertical bar 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 vertical bar U vertical bar parameter subsets and the approximate number of minimal k vertical bar U vertical bar parameter subsets. As an application of k vertical bar U vertical bar 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.