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

This article investigates <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> parameter subsets of a soft set matrix whose column sums are integral multiples of <inline-formula> <tex-math notation="LaTeX">$|U|$ </tex-math></inline-formula> (i.e., the number of objects in the soft set domain <inline-formula> <tex-math notation="LaTeX">$U$ </tex-math></inline-formula>). 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 <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> parameter subsets, whose any proper subset cannot be a <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> parameter subsets and the approximate number of minimal <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> parameter subsets. As an application of <inline-formula> <tex-math notation="LaTeX">$k|U|$ </tex-math></inline-formula> 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.


I. INTRODUCTION
In 1999, Molodtsov introduced the theory of the soft set [1], which is a novel mathematical tool for dealing with uncertainties and vagueness. Many works have been conducted by researchers on soft set theory and its potential applications. Soft set theory has been investigated in combination with algebraic structure [2]- [4], topological structure [5]- [7] and partial order structure [7]- [9], where the abovementioned algebraic structure also includes logic algebraic structure [10]- [12]. Soft set theory is also combined and compared with other mathematical theories designed for modeling various types of vague concepts, such as fuzzy sets [13]- [18], rough sets [19]- [22] and probability theory [24].
The associate editor coordinating the review of this manuscript and approving it for publication was Yizhang Jiang . There is also another important branch of soft set research. Quantitative research work has been performed with respect to the data of the soft set itself. Reference [25] studied a kind of soft set with specific measurement structure, i.e., the bijection soft set. In a bijection soft set, every object belongs to a unique parameter approximation. Reference [26] applied it to the reduction of large-scale high-dimensional information systems. In [27], the parameter mapping subsets are granted the semantics of concept extension. Combining with the three-way decision theories, the structural econometric research of POS, NEG and BND (i.e., positive domain, negative domain and boundary domain) of 0-1 information distribution for a soft set is given. The three-way decision ideas expand the connotation of the soft set and provide new ideas and directions for the study of soft set reduction [27]- [29]. In [30] and [31], the 0-1 or [0,1] interval-valued VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ modeling of the soft set is understood as the valuation in binary or fuzzy logic, respectively. And the approximate reasoning mode in the soft set framework is studied based on quantitative propositional logic [32].
The parameter reduction problems of soft sets actually deal with a kind of structure for the parameter domain. Different kinds of parameter reduction problems [33]- [38] correspond with different criterions. Most of them are required to maintain the same decision making results after a parameter reduction operation. For example, a normal parameter reduction is a parameter subset whose sums of rows are all equal. This means that if they are deleted from the parameter domain, the same choice values will be lost, the same rank of the objects will be obtained. Efforts have been devoted towards issues concerning parameter reduction of soft sets or fuzzy soft sets [34], [35], [37], [38].
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. This kind of parameter subsets was first presented and proven as a necessary condition for normal parameter reduction by [39]. in another word, a k|U | parameter subset is a candidate for a normal parameter reduction of the soft set. In [39] a normal parameter reduction method of soft sets is given by using this property. Since the sum of each column (i.e., the number of elements in the corresponding parameter approximation of the soft set) can be calculated in advance, unnecessary repetition can be avoided from this point of view.
In this article we focus on the minimal k|U | parameter subsets of soft sets. In [40], a hierarchical algorithm for computing all minimal k|U | parameter subsets is studied, and then the normal parameter reduction problem of the soft set can be solved by testing the disjoint combinations of these minimal k|U | parameter subsets. However, the hierarchical algorithm involves a high computing complexity. So 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.
Our idea is as follows. Since the column sum of each parameter (i.e., the number of 1 values) does not consider the specific distribution of the 0 or 1 values, it is possible to classify these parameters into different classes just according to their corresponding column sums. Hence it becomes a question of the integer partition problem. Furthermore, we can conduct integer factorization offline. For a series of soft sets with the same size, we only need to filter the factorization results according to the related constraints induced by the soft set data itself. This is the motivation of our paper. Since the algorithm proposed in [39] didn't give an explicit method for computing all k|U | parameter subsets. That's one possible application of our method. We also intends to generate the integer partition method to the normal parameter reduction problems of soft sets.
The remainder of this article is organized as follows. Section 2 introduces basic concepts such as the soft set, k|U | parameter subsets and the main problems to be investigated with respect to k|U | parameter subsets. The theoretical foundation of this article will be provided in section 3. In section 4, four algorithms including the offline and online algorithm for minimal k|U | parameter subsets will be given. Section 5 will list and analyze the experimental results for the algorithms proposed in section 4. Finally, 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.

II. PRELIMINARIES
In this article, suppose that U = {u 1 , u 2 , · · · , u n } is a finite set of objects, and E is a set of parameters. For example, the attributes in information systems can be taken as parameters. ℘(U ) means the powerset of U , and |A| means the cardinality of set A. By [1] and [41], we have basic concepts about soft sets shown in Definitions 2.1 and 2.2.
means the subset of U corresponding with parameter e. We also use F(u, e) = 1 (F(u, e) = 0) to indicate that u is (not) an element of F(e). Definition 2.2 (Support set of Parameters for Objects): Let S = (F, A) be a soft set over U . ∀u ∈ U , define the support set of parameters for u as the set {e ∈ A|F(u, e) = 1}, denoted by supp(u). We write σ S as σ for short if the underlying soft set S is explicit.
Definition 2.4: Let S = (F, A) be a soft set over U , and B ⊆ A define S F (e) = u∈U F(u, e); S F (B) = e∈B S F (e).

Aims (Problems) to be investigated in this article
• Given an arbitrary soft set S = (F, A) over U , propose a combinatorial formula for computing the number of k|U | parameter subsets K|U|.
• Given an arbitrary soft set S = (F, A) over U , propose a combinatorial formula for computing the approximate number of all minimal k|U | parameter subsets.
• (Main Problem 2.1 in this article ) Given an arbitrary soft set S = (F, A) over U , develop an offline and online algorithm for computing the set of all minimal k|U | parameter subsets K|U| minimal based on the integer partition method.
• Given an arbitrary soft set S = (F, A) over U , investigate the application of k|U | parameter subsets in normal parameter reduction problems of S. Compare our integer partition algorithm with the idea of Ma et al. based on numeration [39].

III. THEORETICAL FOUNDATIONS FOR ALL MINIMAL k|U | PARAMETER SUBSETS OF A SOFT SET
In this section, we first wish to investigate an important question: Since values of S F (e i ) belong to real numbers R, and the relation ''='' on R is an equivalent relation, it is easy to obtain: Proposition 3.1: Let S = (F, A) be a soft set over U , then the relation R on parameters set A is an equivalent relation on A.
With R, we can divide A into different equivalent classes. We use [e i ] R to denote the equivalent class containing e i . That is, [e i ] R = {e ∈ A|eRe i }. We use U to denote the set {0, 1, 2, · · · , |U |}.

C. TRANSFORMATION METHOD FROM INTEGER PARTITIONS TO COMBINATIONS OF PARAMETERS
In this subsection, we will connect the above theories with our problem: that is, how to transform an integer partition to a combination of parameters. This question is not difficult due to the function F(see expression (3.2)).
Obviously, we have

D. COMBINATORIAL CALCULATION FORMULA FOR NUMBER OF k|U | PARAMETER SUBSETS
In this subsection, we will focus on the number of all k|U | partitions satisfying the constraints induced by a soft set S over U . The combinatorial calculation formula for number of k|U | parameter subsets will be proposed. We will also propose an approximate method for the number of minimal k|U | Parameter Subsets. Denote the descending sequence of S F (e i ) as S. Denote S(1), i.e., the maximal element in S, as Max(S).
According to Definition 3.6 and Definition 3.7, we have and and and if such J * does not exist, then • S in P S (M − JK ) denotes the part of S with elements less than or equal to K − 1.
Proof: We prove this by the contrary. If B is in K|U| minimal such that B ← ω 1 , i.e., B is obtained by transforming ω 1 to parameter sets of soft set S, then we see that ω 1 must satisfy the constraint (3.5) with respect to S = (F, A). Therefore, due to the fact that ω 1 − ω 2 is nonnegative, ω 2 must also satisfy the constraint (3.5). Then, B| ω 2 must satisfy Definition 2.5. This is a contradiction since B ← ω 1 and B is minimal.
According to Theorem 3.2, we provide FIGURE 1 as follows. In FIGURE 1, the red way means that we transform all partitions to combinations of parameters, then we conduct the minimalization operation with respect to these combinations. By Theorem 3.2, this is not an optimal way. The green way in FIGURE 1 means that we first execute the minimalization operation to the partitions. As a result, we can reduce the complexity.      where J K means that J is related with K .
We thus determine an iterative formula (3.10) which involves only one function P S (K ). This enables us to compute the number of integer partitions of arbitrarily generated integer M and partition factors sery S.

Algorithm 1 The Number of All
• S in P S (M − JK ) means the part of S with elements t satisfying the following two conditions: (i) t is less than or equal to K − 1.
(ii) t = m − K , i.e., t + K = m. As to the number of all minimal integer partitions, i.e., |k|U| minimal |, it becomes much more complicated. This is the case because we must filter those combinations which cannot be minimal solutions. Based on Algorithm 1, we propose Algorithm 2 which can plane part of nonminimal parameter subsets.

B. OFFLINE AND ONLINE ALGORITHMS FOR ALL MINIMAL k|U | PARAMETER SUBSETS OF A SOFT SET
We have constructed the preliminary foundation and introduced the basic idea for our problem in the above subsections. In this subsection, we will present an offline and online algorithm for all minimal k|U | parameter subsets of a soft set based on integer partition. We separate our algorithm into two stages. One is called the offline stage, in which we attempt to fulfill the common work for soft sets with the same |U |. The other is the online stage, which can handle a series of soft sets with different |A|. See FIGURE 2 and Algorithm 3.

Algorithm 3 An Offline and Online Algorithm for All
Minimal k|U | Parameter Subsets Offline stage: 1: Set up |U |, i.e., the number of objects in the soft sets to be processed. 2: Compute K = |U | − 1. 3: Create the integer partition to k|U | under additional operation with factors as {1, 2, · · · , |U |}, where k = 1, 2, · · · , K . 4: Perform minimal operation on the partitions obtained in Step 3. Online stage: 5: Input soft set S = (F, A) with the same |U |. 6: Compute the sums of columns of the input soft set, and classify the parameter set into different parts [e] R ∈ R generated in Step 6. 7: Filter the minimal partitions obtained in Step 4, which satisfy the constraints of S. 8: Conduct the transformation operation for these partitions with the results obtained in Step 7. 9: Output the final results.

C. ALGORITHMS FOR NORMAL PARAMETER REDUCTION PROBLEMS
In this section, we attempt to implement the idea of the partition into the normal parameter reduction problems of soft sets.
According to [42], we have the following concepts about parameter reduction of soft sets. Note that we do not require the minimality condition for normal parameter reductions as defined in [33].
See FIGURE 3 for a sketch map of Algorithm 4, Inspired by the integer partition idea, we propose Algorithm 4 for normal parameter reduction of soft sets. Its novelty lies in the fact that we first figure out all the integer partions with respect to possible k|U |, where we use the sums of different columns as factors.

V. EXPERIMENTAL RESULTS
• The experiments conducted in this section (i) The combinatorial property for the number of k|U | and approximate number of minimal k|U | parameter subsets. This part involves Algorithm 1, Algorithm 2 and Algorithm 3.
(ii) Experimental results of the Offline and Online Algorithms for all minimal k|U | parameter subsets. This part involves Algorithm 3.
(iii) Comparison of experimental results between Algorithm 4 and Algorithm 5 in solving normal parameter reduction problems. • Equipment and data generation method (i) Our experiments perform on a PC with an AMD Ryzen 5 3500U 2.10GHz CPU, 8GB RAM and the Win10 professional operating system.
(ii) Our data generate in the following way: first, we use the rand function of MATLAB to generate a uniformly distributed matrix of random numbers in the [0,1] interval. Numbers less than or equal to N in the matrix are then changed to 1, and the rest of the numbers are changed to 0. We can thus obtain a matrix with a ratio of 1 to N .   in FIGURE 7. We see that the larger |U | is, the smaller the number is with respect to the same ratio of 1. When the ratio of 1 is equal to 0.2 to 0.5, the average numbers of k|U | parameter subsets are very close with respect to the same |U |.
(iv) Let |U | = 10, and |A| takes values from 8, 10,. . . , 24, where the ratio of 1 is equal to 0.1, 0.2, 0.3, 0.4 and 0.5. We consider the average number of k|U | parameter subsets in 100 iterations of Algorithm 1. The results list in FIGURE 8. We see that the larger |U | is, the larger the number is with respect to the same ratio of 1. When the ratio of 1 is equal to 0.2 to 0.5, the average numbers of k|U | parameter subsets are very close with respect to the same |A|.
(v) Let |A| = 16, and |A| takes values from 8, 10,. . . , 24, where the ratio of 1 is equal to 0.3. |U | takes values from 5, 10, 15, 20, 25. We compare the average number of minimal k|U | parameter subsets in 100 iterations of Algorithm 1, Algorithm 2 and Algorithm 3. The results list in FIGURE 9. We see that there is little change with respect to the exact VOLUME 8, 2020  number of minimal k|U | parameter subsets. As far as our experiments, the approximate number approach the total number of k|U | parameter subsets. We also observe that there is substantial room for improvement of Algorithm 2.
(vi) Let |U | = 15, and |A| takes values from 8, 10,. . . , 18, where the ratio of 1 is equal to 0.3. We compare the average number of minimal k|U | parameter subsets in 100 iterations of Algorithm 1, Algorithm 2 and Algorithm 3. The results list in FIGURE 10. It is shown that the differences among these three numbers increase with the growth of |A|.
(vi) Let |U | = 15 and |A| = 14. The ratio of 1 takes values from 0.1 to 0.5. We compare the average number of minimal k|U | parameter subsets in 100 iterations of Algorithm 1, Algorithm 2 and Algorithm 3. The results list in FIGURE 11. It is shown that the differences between Algorithm 1 and Algorithm 2 became smaller when the ratio of 1 approaches 0.5. The total number of minimal parameter subsets change little when the ratio of 1 is equal to 0.2 to 0.5.  In the offline stage, with k becoming larger, we see that the larger k is, the more possibly the partitial solutions of k|U | will be larger (from the order of inclusion among sets) than those induced by k 1 |U |, where k 1 < k. In other words, for the minimal operation, k|U | contributes no new minimal solutions. Therefore, we can actually execute the partition operation to |U |, 2|U |, · · · , k|U |. The identity of K is very interesting. What is the relationship between K and |U |? In order to answer this question, in this article we choose to conduct experiments. According to TABLE 5, it is shown that K = |U | − 1. In TABLE 6, we list the numerical results of all minimal solutions induced by different k|U |, where k <= |U | − 1.      (ii) Let |U | = 8, |A| = 8, 10, · · · , · · · , 30. In the online stage, we compute the average time cost for expanding operation. The results with respect to different ratios of 1 are shown in FIGURE 13. With respect to the same ratio of 1, the time cost escalates when |A| increase.
(iii) Let |U | = 8, |A| = 20. In the online stage, we compare the average time cost for the filtering and expanding operation. The results with respect to different ratios of 1 are shown in FIGURE 14. It is shown that the time cost for the filtering operation is larger than that of the expanding process.
(iv) Let |U | = 8, |A| = 8, 10, · · · , · · · , 30. We compute the average number of solutions. The results with respect to different ratios of 1 are shown in FIGURE 15. When the ratio of 1 is equal to 0.2, the number is the largest when |A| is larger than 14.
(v) Let |U | = 8, |A| = 20. We compute the average number of solutions. The results with respect to different ratios of 1 are shown in FIGURE 16. When the ratio of 1 is equal to 0.2, the number is the largest.        . It is shown that Algorithm 4 exhibits an obvious advantage over Algorithm 5 with respect to each ratio of 1. For both Algorithm 4 and Algorithm 5, performance do not change much when the ratio grows from 0.1 to 0.5.

VI. CONCLUSION AND FUTURE WORK
This article focuses on the research of k|U | parameter subsets and minimal k|U | parameter subsets. The investigation of such data structures is helpful for information mining and feature extraction of 0-1 information systems.

A. AIMS FULFILLED AND NOVELTY
The main contributions of this article are as follows: (i) We have explored the combinatorial property of k|U | combinations and developed a recursive algorithm for computing the total number of k|U | subsets of parameters.
(ii) Based on (i), we propose a method for computing an upper bound of the minimal number of k|U | subsets of parameters.
(iii) We have proposed an offline and online algorithm for minimal k|U | parameter subsets. Its novelty is based on offline integer partition. We need only to check the partition results according to the related constraints, which are induced by soft set data to come online.
(iv) As an application of k|U | parameter subsets, the method of integer partition has also been extended for normal parameter reduction problems of soft sets.

B. LIMITATIONS OF THE PROPOSED METHOD
(i) We have not yet provided a combinatorial method for computing all k|U | combinations and all minimal k|U | combinations. There is still much room for improving the algorithm: an upper bound of the number of minimal k|U | subsets of parameters requires improvement.
(ii) For the application of our method in normal parameter reduction, since excessive integer partitions exist, the efficiency of the proposed algorithm declines when the size increases (especially with respect to the growth in the number of columns).
(iii) When the number of input soft sets is small, or their scales are different, the benefits produced by offline operation will be offset by the integer partition workload. In this case, the effect of a direct online integer partition based on the constraints of each soft set will be better.
(iv) It should be mentioned that the minimal k|U | partition only considers the information of column sums and does not use the information of row distribution. As a result, minimal k|U | partition cannot guarantee a normal parameter reduction. We must consider all k|U | reductions when we use the integer partition method.
C. FUTURE WORK (i) It is possible that while we compute the number of the k|U | subsets, we can output the exact solutions by recording variables such as [K , J ] in expressions (3.8) and (3.9). This therefore represents potential future work for a real combinatorial method.
(ii) Our work offers a potential way to propose an alternative value for the integers to become partitions. In other words, we need to consider a revised necessary condition for normal parameter reduction of soft sets. The idea of integer partition enables us to work on this direction.
(iii) For the normal parameter reduction problems, the linear programming method in [42] can be combined with the integer partition method proposed in this article. On the one hand, the linear constraints in the linear programming method can constrain the occurrence times of partition factors or the combination of multiple partition factors in the process of integer partition, so the solving process can be simplified. On the other hand, the combination of k|U | constraints and the existing linear constraints in [42] can be used to mine normal parameter reductions with relevant features.
(iv) It is very interesting to combine our theory of 0-1 valued information systems with other kinds of information systems, such as fuzzy-valued information systems and Pythagorean fuzzy-valued information [43]. We also want to explore the potential connections of our algorithms for the normal parameter reduction problems with other fields such as the field of biology [44], [45].
BANGHE HAN received the B.S. degree in mathematics and applied mathematics, the M.S. degree in uncertainty reasoning, and the Ph.D. degree in computational intelligence from Shaanxi Normal University, Xi'an, Shaanxi, China, in 2004China, in , 2007China, in , and 2011 From 2009 to 2015, he was a Lecturer with the School of Mathematics and Statistics, Xidian University, Xi'an, where he has been an Associate Professor, since 2015. His research interests include uncertainty reasoning theories, such as fuzzy sets, fuzzy logic, soft sets, rough sets, and so on. His awards include the First Prize of the Excellent Paper Award for Young People of the Shaanxi Mathematics Association, in 2014, and the Second Prize of the Xi'an Science and Technology Progress Award, in 2017.
RUIZE WU is currently pursuing the degree with the School of Physics and Optoelectronic Engineering, Xidian University.
His research interests include electromagnetic wave propagation and antenna design. He received scientific research training by participating in three undergraduate mathematical contests in modeling during his two years in college and he is participating in his fourth mathematical modeling contest as the Team Leader. He also participated in the China Undergraduate Physics Tournament (CUPT) and was promoted to the regional competition. He has been working on programming problems related with parameter reduction problem soft sets for one year and has a strong interest in mathematical research and mathematical modeling. He likes to discover hidden patterns in mathematical problems.
RUNQING XU was born in Shijiazhuang, China, in 1996. She is currently pursuing the master's degree with Qinghai Normal University.
Her research interest includes intelligent information processing for blockchain. She is quite interested in the scientific research of mathematics, especially for cryptography algorithms. She has been engaged in soft set study for nearly one year and has a strong interest in mathematical research. She enjoys exploring and discovering unknown mathematical laws in mathematical research. VOLUME 8, 2020