The Relationship between the Unicost Set Covering Problem and the Attribute Reduction Problem in Rough Set Theory

1e unicost set covering problem and the attribute reduction problem are NP-complete problems. In this paper, the relationship between these two problems are discussed. Based on the transformability between attribute reductions and minimal solutions in unicost set covering models, two methods are provided. One is to induce an information table from a given unicost set covering model. With no doubt, it shows that the unicost set covering problem can be investigated by rough set theory. 1e other is to induce a unicost set covering model from a given information table. Similarly, it shows that the attribute reduction problem can be studied by set covering theory. As an application of the proposed theoretical results, a rough set heuristic algorithm is presented for the unicost set covering problem.


Introduction
e set covering problem (SCP) is a classical problem in operational research for combination optimization. It is popularly applied in aviation personnel scheduling, circuit design, transport vehicle route arrangement, service location, job assignment in manufacturing, selection of operators, etc. [1][2][3]. e SCP can be described as follows: selecting several columns from a m− row n− column 0 − 1 matrix M � (m ij ) m×n such that these columns can cover all rows with the least cost. e set of selected columns is called a solution of the SCP [4]. Here, (1) choosing each row must pay a certain cost. (2) m ij is the i row and the j column element of M. m ij � 1 means that the column j covers the row i. When all rows have identical cost, for convenience, consider them as 1, the SCP is called the unicost set covering problem (USCP) [5]. is problem is also called the minimum cardinality set covering problem (MCSCP) [6] or the location set covering problem (LSCP) [7]. In this paper, we use the name of the unicost set covering problem. e SCP is a NP-complete problem [8]. Various methods have been proposed to solve it, such as branch and bound complete method [9], genetic method [10], ant colony method [11], and others [12]. Due to the increasing scale of USCP, the time complexity of these methods increases exponentially. Hence, these methods are all effective relatively and there is no efficient method absolutely to handle it, so far.
Rough set theory [13] has been proved to be a useful tool in handling inexact, uncertain, and fuzzy knowledge in information tables [14]. It provides a powerful basis in discovering important data structures and classifying complex objects. Up to now, it has become a hot research area of intelligent computing. As an important concept in rough set theory, the purpose of attribute reduction is to find the minimal attribute set with the same knowledge description ability as the whole attribute set [13]. Scholars have successfully used attribute reduction in data mining [15,16], pattern recognition [17,18], and artificial intelligence and classification [19][20][21] in the past twenty years. e attribute reduction problem (ARP) attracts many researchers to study it [18,20,[22][23][24][25]. Skowron and Rauszer proposed a beautiful method, which is effective to obtain all attribute reductions based on discernibility matrices [20]. e method constructed a discernibility function and showed that the set of all attribute reductions is the set of prime implicants of the discernibility function. However, the time complexity of this method is NP-hard [26]. e cross research of set covering and rough set theory has attracted attention of some scholars. Covering-based rough sets [27] is an import interact way of set covering and rough sets. Zhu and Wang proposed three types of covering-based rough sets [28].
ese models have the ability to implement other features such as multigranularity or fuzziness [29]. Based on 0-1 integer programming, Xu et al. gave an attribute reduction method to deal with the dynamic data [30]. In the variable precision rough set model, Liu et al. provided an approach to calculate the attribute reduction using set covering concepts [31]. In order to deal with the test-cost-sensitive reduction problem [32], Tan et al. presented an optimization algorithm based on set covering theory [33]. Xu et al. proposed an algorithm based on multirelation granular computing model for the USCP [34]. In this paper, we mainly build the relationship between the USCP and the ARP. Firstly, by constructing an induced information table of a given USCP, we find that computing a minimal solution or a minimum solution in a USCP can be converted to computing an attribute reduction or a minimum attribute reduction in the constructed information table. erefore, the USCP can be converted into the APR in rough set theory. Secondly, by constructing an induced unicost set covering (USC) model according to an information table, we find that calculating an attribute reduction or a minimum attribute reduction of an information table is equal to calculating a minimal solution or a minimum solution of the constructed USCP. erefore, the APR can also be converted into the USCP. Let the two problems to be characterized by each other, and bringing new methods mutually is the main aim of the paper. It must be noted that the ARP is a NP-complete problem [26] and the SCP is also a NP-complete problem [8], and using set covering theory to find the optimal attribute reduction with the minimum number of attribute is also NP-hard.
Here, we briefly introduce the contents of the following sections. Section 2 mainly introduces basic concepts of the SCP, the USCP, and the APR in rough set theory, respectively. Section 3 constructs an information table by a USC model and studies the relationship between the USCP and the ARP in the constructed information table. Section 4 constructs a USC model by an information table and considers the relationship between the ARP and the USCP in the constructed USC model. Section 5 provides a heuristic minimal solution method based on rough set theory for the USCP and followed by an example. Section 6 gives the conclusions.

Preliminaries
In this section, some basic concepts of SCP, USCP, information table, and ARP in rough sets are reviewed.

e SCP and the USCP.
In addition to what is described in Section 1, the mathematical formulation of the SCP model is usually described as follows [35]. Let E � e 1 , e 2 , . . . , e m be a set of elements and S � s 1 , s 2 , . . . , s n | ∅ ≠ s j ⊆ E, ∪s j � E} be a subsets collection of E and cover all elements in E, each subset s j associates a cost c j > 0. e pair (E, S) is called a set covering model. S is also called a set covering of E. e SCP is to find a minimal cost set covering S ′ ⊆ S, where all elements in E can be covered and the cost is minimal. e USCP is to find a minimal set covering S ′ ⊆ S, where all elements in E can be covered and any nonempty subsets of S ′ is no longer a set covering of E and S ′ is also call a minimal solution of the USCP. Usually, there are several minimal solutions of a USCP, and the one with the minimum cardinal number is called the minimum solution.
e SCP also can be described by the following 0 − 1 integer programming [4]: Here, m ij is the i row and the j column element of the 0 − 1 matrix M. m ij � 1 means that the j column covers the i row; s j flags whether the j column is included in the solution.
at is, when s j takes 1, it means j column is included in the solution. c j represents the cost value of j column. Formula (1) denotes the minimal cost required for the solution; formula (2) implies that the solution must cover all lines. When formulas (1) and (2) are satisfied, let S ′ � (s 1 , s 2 , . . . , s j ) t , where symbol t denotes transposition; then, S ′ is a solution of the SCP. When all subsets in S have identical cost, for convenience, consider them as 1, and this problem is the USCP. at is, in a USCP 0 − 1 integer programming, formula (1) is replaced by the following formula: Here, formulas (2) and (4) imply that a solution of the USCP must cover all lines and the number of the elements in the solution must be least.
Denote USCP by USCP � (E, S), where E � e 1 , e 2 , . . . , e m is the object set and S � s 1 , s 2 , . . . , s n | ∅ ≠ s j ⊆ E, ∪s j � E is a set covering of E with the 0 − 1 matrix M � (m ij ) m×n .
Note. e theory of soft sets introduced by Molodtsov [36] is a relatively new approach to discuss vagueness and getting popularity among the researchers. N-soft set, first introduced by Fatimah et al. [37], can provide a finer granular structure with higher distinguishable power [38]. Given [39]. e set covering model (E, S) is a special N-soft set, where ∀e j ∈ S, e j ≠ ∅ and ∪S � E. Similarly, the USC model is a special soft set. Example 1. In Ad-hoc sensor networks, the signal coverage is the main measure of network quality of service. Under the premise of ensuring the quality of signal coverage, how to save network construction cost as much as possible is particularly important. Let Ad-hoc sensor networks be as shown in Figure 1. e five circular areas s 1 , s 2 , . . . , s 5 are the signal coverage of the same type of sensors. e 1 , e 2 , . . . , e 9 are nine customers, who want to receive the networks service. How to finding the optimum scheme for setting up sensors?
Consider nine service customers as objects. en, the object set E � e 1 , e 2 , . . .
Because five sensors are the same type, Example 1 is a USCP. A mininal solution of the USCP is an optimum sensors setting scheme for Example 1. Example 1 can be described by the following 0 − 1 integer programming: Use Matlab2016a to run the following procedures: We can obtain an optimum solution X � (0, 1, 1, 0, 0) t . at is, s 2 , s 3 is the optimum scheme for setting up sensors in Example 1.
In this paper, we describe Example 1 as a USCP � (E, S), where E � e 1 , e 2 , . . . , e 9 is the object set and

ARP in Rough Set eory.
Rough set theory is one of the most successful tools for uncertainty management. In rough sets, information is presented by an information table or an information system. An information table is usually described as a pair IT � (U, A) with a mapping f from U × A to V � ∪ a∈A V a , where U, called the universe, is a finite and nonempty set of objects, A is a finite and nonempty set of Mathematical Problems in Engineering attributes, and V a , called the a ′ domains, is the value set of attribute a [13].
For any nonempty attribute subset B ⊆ A, where R B is called an indiscernibility relation on U w.r.t. B. It is easy to see that R B is an equivalence relation on the universe U with the partition Attribute reduction is one of the core problems in rough sets. It is a key process for discovering important data structures and classifying complex objects. Based on the indiscernibility relation discussed above, Pawlak proposed the concept of attribute reduction in an information table.
e attribute reduction is a minimal subset of attributes, which has the same indiscernibility relation as R A [40].
If R B � R A , then we call B an attribute consistent set of IT. If B is an attribute consistent set, for ∀C ⊂ B, C is not a consistent set; then, we call B an attribute reduction of the IT.
Definition 1 means that an attribute reduction is a minimal subset of attributes with R B � R A .
Normally, there is more than one attribute reduction in an information table. e one with the minimum cardinal number is called the minimum attribute reduction. RED(IT) denotes the set of all attribute reductions of an information table IT.
From the viewpoint of the importance of attribute, Skowron and Rauszer [20] define the notion of core.
Definition 2 (see [20]). Let IT � (U, A) be an information { } , otherwise a is indispensable. e set of all indispensable attributes is called the core set of IT, denoted as CORE(IT). CORE(IT) is also the intersection of all attribute reductions of IT.
Scholars provided many methods for attribute reduction, where the method based on the discernibility matrix and discernibility function can compute all the attribute reductions of an information table using logical operations [20].
It is usually to consider only the lower triangle or the upper triangle of the matrix. D is the simplified form of D. For Theorem 1 (see [20]).
Skowron and Rauszer [20] proposed a method to compute all the attribute reductions of an information table, which is based on notions of discernibility function and logical operation. A discernibility function f S for an IT is a Boolean function, which has m Boolean variables a 1 , a 2 , . . . , a m corresponding to m attributes a 1 , a 2 , . . . , a m , respectively.
Theorem 3 (see [20]). Let IT � (U, A) be an information table and f S (a 1 , a 2 , . . . , a m ) be the discernibility function. An eorem 3 means that a prime implicant of discernibility function f S (a 1 , a 2 , . . . , a m ) is an attribute reduction of IT, and all the prime implicants of discernibility function f S (a 1 , a 2 , . . . , a m ) are all the attribute reductions of IT. So, we can use the disjunction ∨ and conjunction ∧ operations to compute all the attribute reductions of an information table. If , where ∧ s r k�1 a k , r ≤ t are all the prime implicants of the discernibility function f S , then B r � a k | k ≤ s r , and r ≤ t is an attribute reduction of IT. Denote all the attribute reductions of IT by B � B r | r ≤ t .
From the above conclusion, one can obtain all the attribute reductions of an information table by computing the discernibility function using Boolean operations. Table 1 1 , a 2 , a 3 , a 4 , a 5 . Table 2 represents the discernibility matrix D of IT, where we only consider the upper triangle matrix.

Example 2. Let IT � (U, A) be an information table, as shown in
e discernibility function f S of IT is Using Boolean logical operations, we can obtain According to eorem 3, IT has two attribute reductions: (a 1 , a 2 , a 4 , a 5 ) and (a 2 , a 3 , a 4 , a 5 ). CORE(IT) � a 2 , a 4 , a 5 .

Inducing an Information Table from a USC Model
In this section, we induce an information table from a USC model and give an example to explain the inducing process. en, we discuss the relationship between the USCP and the ARP in the induced information table.
e induced information table IT � (U, A), constructed by a USC model as above, is a special N-soft set, where ∀e i ∈ U, F(e i , s j ) � 0 or i(1 ≤ i ≤ m, 1 ≤ j ≤ n), and F(e m+1 , s j ) � 0(1 ≤ j ≤ n).

Example 3.
e induced IT of the USCP in Example 1 is shown in Table 3.
(2)⇐ Suppose S ′ ⊆ S is an attribute consist set of the induced IT.
en, for ∀(e i , e k ) ∈ U × U, i ≠ k, d(e i , e k ) ≠ ∅, S ′ ∩ d(e i , e k ) ≠ ∅ ∀e i ∈ E, d(e i , e m + 1) ≠ ∅, and S ′ ∩ d (e i , e m + 1) ≠ ∅ because d(e i , e m + 1) � s j | m ij � 1, 1 ≤ j ≤ n}, there is at least one s j ∈ S ′ and s j ∈ d(e i , e m + 1), thus m ij � 1, s j covers the element e i , and one gets that S ′ is a set covering of the USCP.
Hence, combining (a) and (b) with (c), if s j ∈ S, then s j is a core of the induced IT, ∃e i ∈ E, m ij � 1 and ∀s t ∈ S, s t ≠ s j , m it � 0.
Definition 5 and eorems 6 and 7 make it possible to use attribute reduction methods to compute a minimal solution or the minimum solution of a USCP. us, the USCP can be solved by using rough set theory.
In the following example, we illustrate how to compute a minimal solution or the minimum solution of a USCP by calculating an attribute reduction or the minimum attribute reduction of the constructed information table.
□ Table 2: e discernibility matrix D of IS in Example 2. a 3 , a 5  a 1 , a 2 , a 3  a 1 , a 2 , a 3 , a 5  a 1 , a 2 , a 3 , a 4    e discernibility matrix D of IT constructed by the USCP in Example 1 can be represented in Table 4, where we only consider the upper triangle matrix.
By Table 4, the discernibility function f S of the induced IT is as follows: Using Boolean logical operations, we can obtain  In what follows, with the help of the simplified discernibility set proposed by Yao and Zhao [18], we construct a USCP model from a given information table, and the ARP is corresponding to the USCP. us, we give an example to explain the constructing process. en, we discuss the connection between the ARP and the constructed USCP.

Inducing a USC Model from an Information
Definition 6 (see [18]). For a matrix element d( ′ , x j ′ ) absorb and replace another element d(x i , x j ), and we call the process element absorption.
Definition 7 (see [18]). For a discernibility matrix D � (d(x i , x j )) n×n , if we use element absorption to process all the elements in it, and then it can be converted into another simplified discernibility matrix D ′ � (d ′ (x i , x j )) n×n . We call the process matrix absorption and call D ′ the simplified discernibility matrix of D.
In Definition 7, we can see that no elements in D ′ can be absorbed by each other. Yao and Zhao point out that the discernibility matrix D ′ has the same set of attribute reductions as the discernibility matrix D [18], so do D ′ and D.

Example 5.
e simplified discernibility matrix D ′ of D in Example 2 can be represented in Table 5 as follows, where we only consider the upper triangle matrix.
Using the simplified discernibility matrix D ′ , we can obtain f S a 1 , a 2 , a 3 , a 4 , a 5 Using Boolean logical operations, we have f S a 1 , a 2 , a 3 , a 4 e result is the same as Example 2. Inspired by the result proposed by [18], in the following, we construct an induced USC model from a given information table by using the simplified discernibility matrix.
Definition 8. Given an information table IT � (U, A), U � x 1 , x 2 , . . . , x m , A � a 1 , a 2 , . . . , a n , and D ′ � d 1 , d 2 , . . . , d t }: (1) Assign D ′ as an object set E, that is, E � d 1 , d 2 , . . . , d t } (2) Assign a covering set S � s a 1 , s a 2 , . . . , s a n with s a j � d i | a j ∈ d i , 1 ≤ i ≤ t , 1 ≤ j ≤ n e object set E and the covering set S constitute a USC model with the t− row n− column 0 − 1 matrix M � (m ij ) t×n as follows: S) is called the induced USC model of the given IT.
In Definition 8, for ∀d i ∈ E, d i ≠ ∅, there is at least ∃a j ∈ A, a j ∈ d i , such that d i ∈ s a j and d i ∈ ∪ n j�1 S. us, we have ∪ n j�1 S � E, i.e., S is exactly a set covering of E. Hence, the pair (E, S) is a USCP.

Example 6.
Construct an induced USCP model from the IT in Example 2.
Theorem 9. Given an information table IT � (U, A), a 2 , . . . , a n , and Proof of eorem 9. (1)⇒ Suppose B is an consistent set of the given IT. en, for ∀d i ∈ D ′ , B ∩ d i ≠ ∅, that is, there is at least ∃a j ∈ B and a j ∈ d i , according to Definition 8, so d i ∈ s a j , d i is covered by s a j . Hence, S ′ � s a j | a j ∈ B covers d i . us, S ′ � s a j | a j ∈ B is a set covering of the induced USCP.
(2)⇐ Suppose S ′ � s a j | a j ∈ B is a set covering of the induced USCP.
en, for ∀d i ∈ E, ∃s j ∈ S ′ , d i ∈ s a j , according to Definition 8, a j ∈ d i , so a j ∩ d i ≠ ∅. Hence, B ∩ d i ≠ ∅, B is an attribute consistent set of the given IT.
Combining (1) with (2), it completes the proof. Similar to Section 3, we can easily obtain some results. Proof of eorem 11. By eorem 10, it is easy to prove. □

Theorem 12. Given an information table IT
A � a 1 , a 2 , . . . , a n , and . . , s n , and the 0 − 1 matrix M � (m ij ) t×n . For an attribute a j ∈ A, a j is a core of the given IT iff ∃d i ∈ E, d i � a j in the induced USCP.
Proof of eorem 12. (1)⇒ Suppose a j ∈ A, a j is a core of the given IT.
en, ∃d i ∈ D ′ , d i � a j . According to Definition 8, d i ∈ E, ∃d i ∈ E, d i � a j in the induced USCP.
(2)⇐ Suppose ∃d i ∈ E, d i � a j in the induced USCP, according to Definition 8, d i � a j ∈ D ′ , by eorem 2, we have that a j is a core of the given IT.
Combining (1) and (2), it completes the proof. From eorems 10 and 11, we can see that an attribute reduction or a minimum attribute reduction of a given information table is a solution or the minimum solution of the constructed USCP. us, the ARP can be converted into the USCP.
We can obtain all of the minimal solution of the constructed USCP in Example 6 by using Matlab: X � (1, 1, 0, 1, 1) t and (0, 1, 1, 1, 1) t . at is, a 1 , a 2 , a 4 , a 5 and a 2 , a 3 , a 4 , a 5 are all attribute reductions of the information table IT in Example 2. And by eorem 12, a 2 , a 4 , a 5 is the core set.

A Rough Set Heuristic Algorithm for USCP
Based on the discussion above, the relationship between the USCP and the ARP is established. One can investigate the USCP by rough set theory and discuss the ARP of an information table via set covering theory.
In this section, based on rough set theory, we propose a heuristic algorithm based on rough set theory for USCP as an application of the proposed theoretical results.
ere is no doubt that the method based on discernibility function can compute all attribute reductions or an minimum attribute reduction, but the time complexity of it is NP-hard [26]. However, finding an attribute reduction based on heuristic algorithms can dramatically reduce the time complexity. By eliminating irrelevant states or unlikely possibilities, heuristic algorithms can lessen the computational efficiently. e classical heuristic algorithms are discernibility matrix methods [41][42][43], positive-region methods Note. e simplified discernibility matrix D ′ is not the unique one simplified discernibility matrix of D. e ordering in which different elements are absorbed can obtain different simplified discernibility matrix. Nevertheless, all the absorbed matrices have the same set of attribute reductions as the discernibility matrix D [18]. [44][45][46], and information entropy methods [23,47,48]. Information entropy methods usually use entropy to measure the importance of knowledge. ey usually make the entropy as a heuristic function and select the more important knowledge based on the entropy gradually to construct an attribute reduction. e notion of entropy is introduced by Shannon, and Slezak applied Shannon's entropy to compute an attribute reduction in information table [23].
Definition 9. (see [23]). Given an information table as IT � (U, A). ∀B ⊆ A and where H(B) is called the entropy of B.
Here, X i ∈ (U/R B ) � X 1 , X 2 , . . . , X r , and | · | means the number of elements in a set. If H(B) � H(A), then B is an attribute reduction of IT [23,47]. e significance of attribute set B can be measured by H(B).
Definition 10 (see [23]). Given an information table IT � (U, A). B ⊆ A, ∀a ∉ B, and where Sig(a, B) is called the significance of attribute a w.r.t. B. Now, we construct a rough set heuristic algorithm for USCP via the entropy.
By Algorithm 1, we can compute a minimal solution of a given USCP by selecting the most important attribute one by one according to its entropy.
In Algorithm 1, step 2 selects all cores according to eorem 8 and its time complexity is O (mn). e time complexity of step 3 is O (mn).
Step 4 is the most important step. It selects the most import attribute in S − Red w.r.t. the Red. Its time complexity is O (|U| 2 |A|) [46], that is, O (m 2 n). So, the time complexity of Algorithm 1 is O (m 2 n). (2) For all 1 ≤ i ≤ m, there is not a 1 ≤ j ≤ n, m ij � 1 and ∀1 ≤ t ≤ n, t ≠ j, m it � 0, so Red � ∅. e minimal solution s 2 , s 3 computed by Algorithm 1 is the same as the result computed by Matlab and computed by discernibility function.

Conclusions
is paper has established the relationship between the USCP and the ARP. e results have shown that finding a Input: A USCP � (E, S) with E � e 1 , e 2 , . . . , e m , S � s 1 , s 2 , . . . , s n , M � (m ij ) m×n ; Output: A minimal solution.
(1) Let Red � ∅; (2) For all 1 ≤ i ≤ m, if ∃ 1 ≤ j ≤ n, m ij � 1 and ∀1 ≤ t ≤ n, t ≠ j, m it � 0, then let Red � Red∪ s j ; (3) Induce an information system IS � (U, S) from the given USCP; minimal or a minimum solution of a USC model is equivalent to finding an attribute reduction or a minimum attribute reduction in the induced information table. Also, finding an attribute reduction or a minimum attribute reduction of an information table is equivalent to finding a minimal or a minimum solution in the induced USC model. e results have provided the two problems to be characterized by each other and bring new methods mutually. As an application, a rough set heuristic algorithm for USCP has been given.
In practical applications, data is always not free. In view of this situation, Min et al. formally defined test-cost-sensitive decision system [49]. In test-cost-sensitive decision system, there is a test cost for each data item. e test-costsensitive ARP is to find a reduction which has the minimal cost [32]. In addition, in the SCP, each subset s j associates a cost c j > 0. e SCP is to find a minimal cost set covering where the cost is minimal. Our future work will plan to discuss the relationship between the test-cost-sensitive ARP and the SCP. In this paper, the induced information table constructed by a USC model is an N-soft set. Parameter reduction is an active area of research in soft set theory as well [50]. Discussing the relationship between the parameter reduction problem and the SCP is also a future work of us. In rough set theory and granular computing, one normally considers measures of granularity, as discussed in Section 5. It might be interesting to look at other measures, for example, complexity measures [51]. In the future research, we will study the attribute reduction problem based on complexity measure.

Data Availability
All data, models, and code generated or used during the study appear in the article and all reference data has been annotation references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.