Matching Index of Uncertain Graph: Concept and Algorithm

Transcription

1 Matching Index of Uncertain Graph: Concept and Algorithm Bo Zhang, Jin Peng 2, School of Mathematics and Statistics, Huazhong Normal University Hubei , China 2 Institute of Uncertain Systems, Huanggang Normal University Hubei , China Abstract: In practical applications of graph theory, there is no doubt that some uncertain factors may appear in graphs. This paper employs the uncertainty theory to deal with uncertain factors in uncertain graph. Matching index and perfect matching index of uncertain graph are proposed. Some properties of the matching index are discussed. Furthermore, we give an algorithm to calculate the matching index of uncertain graph. Keywords: Uncertainty theory; Uncertain measure; Matching index; Perfect matching index; Uncertain graph Introduction Since originated in seven bridges problem by Euler in 736, the graph theory has received great attention. For instance, Bondy and Murty [2], Harary [9], Tutte [22], and Xu [23] have done much work in the field of graph theory. In classic graph theory, the edges and the vertices are all deterministic. However, in practical application, as the system becomes more complex, indeterminacy factors may appear in graphs. If the weight of each edge is of indeterminacy, it is not suitable to employ the traditional methods to study the weighted matching problem. Sometimes, whether two vertices are joined by an edge cannot be completely determined in a graph. As a result, the traditional methods can not be used to verify some properties of the graph. In order to deal with indeterminacy factors in graph, random graphs were first defined by Erdös and Rényi [4, 5]. They thought that whether two vertices are joined can be described as a random variable. Since then, the random graph has been studied by many researches, such as Bollobás [], Gilbert [8], Mahmoud et al. [9], etc. Unfortunately, it is not suitable to employ the probability theory to deal with every indeterminacy phenomenon. A fundamental premise of applying probability theory is that the sample size is large enough, and the estimated probability is close enough to the real frequency. However, in the real life, we are frequently lack of observed data about the unknown state of nature, not only for technical difficulties, Corresponding author. Tel: Fax: addresses: pengjin0@tsinghua.org.cn.

2 but also for economic reasons. Then, how does one deal with this kind of indeterminacy factors? Usually, we have to invite some domain experts to evaluate the belief degree that each event will occur. Since human beings usually overweight unlikely events (Kahneman and Tversky [0]), the belief degree has much larger variance than the real frequency. In this case, the probability theory is no longer valid. In order to deal with this indeterminacy phenomenon distinguishing from randomness, Liu [2] founded the uncertainty theory in Up to now, the uncertainty theory has become a branch of mathematics for modeling human uncertainty. In order to describe dynamic uncertain systems, Liu [3] introduced uncertain process. Besides, uncertain calculus was initialized by Liu [4] to deal with differentiation and integration of functions of uncertain processes. Furthermore, You [24] proved some convergence theorems of uncertain sequences, Liu and Xu [8] studied some properties on uncertain variables, and Peng and Iwamura [2] given a sufficient and necessary condition of uncertainty distribution. In practical aspect, the uncertainty theory was first introduced into finance by Liu [4]. In addition, Peng and Yao [20] proposed an uncertain stock model and given the corresponding option pricing formulas. Furthermore, Gao [7] employed the uncertainty theory to investigate the shortest path problem with uncertain arc lengths. Zhang and Peng [25] presented some uncertain programming models for Chinese postman problem in uncertain environment. For exploring the recent developments of the uncertainty theory, the readers may consult Liu [7]. Gao and Gao [6] first defined the uncertain graph. As a powerful tool for modeling human uncertainty, the uncertainty theory is employed to deal with uncertain factors in graph. Recently, Zhang and Peng [26] given an Euler index to show how likely an uncertain graph is Eulerian. An uncertain graph refers to the graph in which whether two vertices of the graph are joined by an edge cannot be completely determined. Then how to verify the properties of the graph? Matching is one of the basic concept of the graph theory. For a given uncertain graph, at how much belief degree we can regard the uncertain graph has a maximum matching? In this paper, the concepts of matching index and perfect matching index of uncertain graph are firstly proposed. With the framework of uncertainty theory, an algorithm to calculate matching index of uncertain graph is given. The remainder of this paper is organized as follows. After introducing some basic concepts and properties of uncertainty theory and graph theory in Section 2, Section 3 proposes the concepts of matching index and perfect matching index of uncertain graph, and also investigates some properties of the matching index. In Section 4, an algorithm to calculate the matching index of uncertain graph is given. The last section concludes this paper with a brief summary. 2 Preliminaries 2. Uncertainty Theory Now, we present some preliminaries from uncertainty theory. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ L is called an event. The set function M{Λ} is called an uncertain measure if it satisfies the following three axioms (Liu [2]): 2

3 ()(Normality Axiom) M{Γ} = ; (2)(Duality Axiom) M{Λ} + M{Λ c } = for any Λ L; (3)(Subadditivity Axiom) For every countable sequence of events {Λ i }, we have { } M Λ i M{Λ i }. i= Uncertain measure is one of the basic concept of the uncertainty theory that is used to indicate the belief degree that an uncertain event may occur. The triplet (Γ, L, M) is called an uncertainty space. In order to obtain an uncertain measure of compound event, the fourth axiom called product axiom was presented by Liu [4]. (4)(Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k =, 2, The product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } where Λ k are arbitrarily chosen events from L k for k =, 2,, respectively. k= i= k= An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers. Liu [2] introduced the independence concept of uncertain variables. The uncertain variables ξ, ξ 2,, ξ n are said to be independent if { n } M (ξ i B i ) = for any Borel sets B, B 2,, B n of R. i= n M{ξ i B i } A function is said to be Boolean if it is a mapping from {0, } n to {0, }. An uncertain variable is said to be Boolean if it takes values either 0 or. A Boolean function f(b, B 2,, {0},, B n ) is said to be an increasing Boolean function if f(b, B 2,, {0},, B n ) = = f(b, B 2,, {0, },, B n ) =. Theorem (Liu [6]) Assume that ξ, ξ 2,, ξ n are independent Boolean uncertain variables, i.e.,, with uncertain measure α i ξ i = 0, with uncertain measure α i for i =, 2,, n. If f is a Boolean function, then ξ = f(ξ, ξ 2,, ξ n ) is a Boolean uncertain variable such that sup M{ξ = } = sup f(x,x 2,,x n)= f(x,x 2,,x n)=0 i= min ν i(x i ), if sup i n f(x,x 2,,x n)= min ν i(x i ), if sup i n where x i take values either 0 or, and ν i are defined by α i, if x i = ν i (x i ) = α i, if x i = 0 for i =, 2,, n, respectively. f(x,x 2,,x n)= min ν i(x i ) < 0.5 i n min ν i(x i ) 0.5 i n 3

4 In more detail, if f is an increasing Boolean function, we have the following theorem. Theorem 2 (Gao and Gao [6]) Assume that ξ, ξ 2,, ξ n are independent Boolean uncertain variables, i.e.,, with uncertain measure α i ξ i = 0, with uncertain measure α i for i =, 2,, n. If f is an increasing Boolean function, then ξ = f(ξ, ξ 2,, ξ n ) is a Boolean uncertain variable such that M{ξ = } = where B i are subsets of {0, }, i =, 2,, n. sup f(b,b 2,,B n)= min M{ξ i B i }, i n 2.2 Matching In this subsection, some basic concepts and terminology of graph theory are introduced, which are come from Bondy and Murty [2]. Definition (Bondy and Murty [2]) A graph G is an order triple (V (G), E(G), ψ G ) consisting of a nonempty vertex set V (G), a set E(G) of edges, and an incidence function ψ G that associates with each edge an unordered pair of (not necessarily distinct) vertices of G. A loop is an edge whose ends are equal. A link is an edge with distinct ends. A simple graph is a graph having no loops and no two of its links join the same pair of vertices. Graph G is finite if both its vertex set and edge set are finite. Every graph in this paper is simple graph. Let G be a graph with vertex set V (G) = {v, v 2,, v n } and edge set E(G) = {e, e 2,, e m }. Then the adjacency matrix of G is the n n matrix x x 2 x n x 2 x 22 x 2n X = x n x n2 x nn where x ij =, if there exists an edge between vertices v i and v j 0, otherwise. Obviously, x ii = 0 and x ij = x ij for i, j =, 2,, n, respectively, that is to say, X is a symmetric matrix. A subset P of E(G) is called a matching in graph G if no two of elements of P are adjacent in G. If some edge of P is incident with vertex v, then v is said to be P -saturated; Otherwise, v is P -unsaturated. A matching P is perfect if every vertex of G is P -saturated; and a matching P is maximum if P P for any matching P. It is clear that a perfect matching must be a maximum matching, but a maximum matching does not necessarily to be a perfect matching. 4

5 3 Matching Index 3. Concepts In classic graph theory, all the edges and vertices of a graph are deterministic, either existing or not. However, as the system becomes more complex, some indeterminacy factors will appear in graphs. In this paper, the indeterminacy factor is that we are not sure whether an edge exists between two vertices. If there is enough historical data to estimate the distribution via statistics, then we can employ random variable to describe this indeterminacy factor. However, if we are lack of history data, or history data is invalid because of unexpected events have occurred. Usually, we may ask some domain experts to give a belief degree that the edge exists. As we stated before, this expert data is just the subject of the uncertainty theory. In 20, Gao and Gao [6] employed the uncertainty theory to graph theory, and proposed the concept of uncertain graph in which all edges are independent and exist with some belief degrees in uncertain measure. The number of vertices in G is called the order of G, while the number of uncertain edges is called its size. Definition 2 (Gao and Gao [6]) A graph of order n is said to be uncertain if it has uncertain adjacency matrix α α 2 α n α 2 α 22 α 2n A = α n α n2 α nn where α ij represent that the edges between vertices v i and v j exist with uncertain measures α ij, i, j =, 2,, n, respectively. Note that α ii = 0 and α ij = α ji for i, j =, 2,, n, respectively, which means A is a symmetric matrix. According to the definition of uncertain graph that the elements of uncertain adjacency matrix are no longer 0 or but numbers in [0, ]. For example, α ij = 0.4 indicates the edge between vertices v i and v j exists with uncertain measure 0.4 and dose not exist with uncertain measure. Different from that in a classical graph, the edge set of uncertain graph G is a set of Boolean uncertain variable E(G) = {ξ 2, ξ 3,, ξ n, ξ 23,, ξ 2n,, ξ (n )n }, where M{ξ ij = } = α ij and M{ξ ij = 0} = α ij, for i < j n. Remove the edges ξ ij satisfying M{ξ ij = } = 0, and we denote the edge set E(G) = {ξ, ξ 2,, ξ m }. Definition 3 Let G be an uncertain graph with edge set E(G) = {ξ, ξ 2,, ξ m }. The underlying graph of G, denoted by G, is a graph obtained from G by replacing each edge by M{ξ ij = } =. In order to analyze graphs and their properties with uncertain factor, we give the following fundamental concept. 5

6 Definition 4 Let G be an uncertain graph with edge set E(G) = {ξ, ξ 2,, ξ m }, and P be a maximum matching of G. The matching function of G is denoted as:, if there is a matching P in graph G such that P = P P (E(G)) = 0, otherwise. Clearly, P (E(G)) is an increasing Boolean function. Now, matching index and perfect matching index are defined below. Definition 5 Let G be an uncertain graph with underlying graph G, and P be a maximum matching of G. The matching index κ(g) of the graph G is the uncertain measure that G has a matching P such that P = P. Definition 6 Let G be an uncertain graph with underlying graph G, and P be a perfect matching of G. The perfect matching index pκ(g) of the graph G is the uncertain measure that G has a matching P such that P = P. Remark An uncertain graph must have a matching index, but it is not true that every uncertain graph has a perfect matching index. Remark 2 An uncertain graph G has a perfect matching index if and only if its underlying graph G has a perfect matching. When an uncertain graph G has a perfect matching index, then the perfect matching index of G is equal to the matching index of G. 3.2 Properties Theorem 3 Assume G is an uncertain graph of order n and its uncertain adjacency matrix is α α 2 α n α 2 α 22 α 2n A =, α n α n2 α nn where α ii = 0 and α ij = α ji for i, j =, 2,, n, respectively. If all edges are independent, then the matching index of G is sup min ν ij(x), P (X)= i<j n κ(g) = sup min ν ij(x), i<j n P (X)=0 if sup P (X)= if sup P (X)= min ν ij(x) < 0.5 i<j n min ν ij(x) 0.5 i<j n where X = x x 2 x n x 2 x 22 x 2n......, x n x n2 x nn 6

7 x ij {0, }, x ij = x ji, x ii = 0, and ν ij are defined by α ij, if x ij = ν ij (X) = α ij, if x ij = 0 for i, j =, 2,, n, respectively, and, if there is a matching P in X such that P = P P (X) = 0, otherwise, where P is the maximum matching of the underlying graph G of graph G. Proof It is clear that G has a matching P such that P = P if and only if P (X) =. Note that P (X) is a Boolean function. It follows from Theorem that the theorem is proved. Example Figure illustrates the uncertain graph G of order 3 size 3 and its uncertain adjacency matrix Figure : Uncertain graph G and its uncertain adjacency matrix Assume P is the maximum matching of the underlying graph G of the uncertain graph G in Figure. It is clear that P =. Since the size of the uncertain graph is 3, its adjacency matrix breaks down into = 8 cases. Assume the adjacency matrix X of G is one of the following seven matrices , 0 0, 0, , 0 0, 0 0 0, Then the graph G has a matching P such that P =, i.e., P (X) = and Assume X is sup P (X)= min ν ij(x) =. i<j

8 Then the graph G does not have a matching P such that P =, i.e., P (X) = 0 and sup P (X)=0 min ν ij(x) = 0.. i<j 3 It follows from Theorem 3 that the matching index of G is κ(g) = sup P (X)=0 min ν ij(x) = 0.9. i<j 3 In more detail, since P (E(G)) is an increasing Boolean function, Theorem 2 directly leads to Theorem 4 Let G be an uncertain graph with edge set E(G) = {ξ, ξ 2,, ξ m }. The matching index of G is κ(g) = sup P (E(G))= min M{ξ i B i } i m where B i are subsets of {0, }, i =, 2,, m. Remark 3 The value of κ(g) (or pκ(g)) of uncertain graph G represents the belief degree that the graph G has a maximum matching (or perfect matching). If κ(g) = (or pκ(g) = ), then the graph G has a maximum matching (or perfect matching) completely; If κ(g) = 0 (or pκ(g) = 0), then the graph G does not have a maximum matching (or perfect matching) at all. Thus the larger the value of κ(g) (or pκ(g)) is, the more true the graph G has a maximum matching (or perfect matching). Remark 4 If the graph G has a maximum matching (or perfect matching) with belief degree α, then the graph G does not have a maximum matching (or perfect matching) with belief degree α. This fact follows from the duality property of uncertain measure. In other words, if κ(g) = α, then M{P (E(G)) } = M{P (E(G)) = } = α. Remark 5 When an uncertain graph G does not have a perfect matching index, we denote pκ(g) = 0 which is reasonable. Moreover, according to Theorem 4, we can easily obtain Corollary Let G be an uncertain graph with edge set E(G) = {ξ, ξ 2,, ξ m }. If no two edges are adjacent in G. Then the matching index of G is the smallest value of M{ξ i = }, i =, 2,, m. Next, we will give another method to calculus the matching index for other more complex uncertain graph. First, we introduce the maximum index spanning matching. Definition 7 Assume that G is an uncertain graph with underlying graph G, and P is a maximum matching of G. A subset P of edge set E(G) is called spanning matching of G if no two of elements of P are adjacent in G, and P = P. The maximum index spanning matching is a spanning matching with maximum matching index. In the following, we will show that the matching index of G is equal to the matching index of the maximum index spanning matching of G. 8

9 Theorem 5 Let G be an uncertain graph with vertex set V (G) = {v, v 2,, v n } and edge set E(G) = {ξ, ξ 2,, ξ m }, and P be the maximum index spanning matching of G. Then we have κ(g) = κ(p ). Proof Clearly, κ(g) κ(p ). Next, we only need to prove κ(g) κ(p ), where κ(g) > 0. Since P (E(G)) is an increasing Boolean function, there must exist a series of {B i }m i=, taking values of {} or {0, }, satisfying κ(g) = sup P (E(G))= min M{ξ i B i } = min M{ξ i B i} = min M{ξ i k = } > 0, () i m i m k t where t is a positive number and subseries {B i k } t k= takes values of {}. Obviously, t P. In fact, we can choose {B i k } t k= such that t = P. If t = P, we have done. If t > P, the subgraph S with edge set E(S) = {ξ i, ξ i2,, ξ it } must contain at least a spanning matching and some other edges, the set of these edges is denoted by A. Choose any edge of A, such as ξ it, and remove it. Then we obtain a new subgraph S with edge set E(S ) = {ξ i, ξ i2,, ξ it }. Furthermore, S must contain a spanning matching and κ(g) κ(s ) min M{ξ i k = } min M{ξ i k = }. (2) k t k t Combining with () and (2), we have κ(g) = κ(s ). Repeating this argument until there is only a spanning matching. That is, we obtain a spanning matching P such that κ(g) = κ(p ). Since P is the maximum index spanning matching, thus κ(g) = κ(p ) κ(p ). This completes the proof. Example 2 Assume that G is an uncertain graph (see Figure 2()). Obviously, there are five spanning matchings of the uncertain graph G (see Figures (2)-(6) in Figure 2). According to Corollary, we obtain the matching index of the spanning matchings shown in Figure 2 as follows: κ(p ) =, κ(p 2 ) =, κ(p 3 ) = 0.7, κ(p 4 ) = 0.8, κ(p 5 ) =. According to Theorem 5, the matching index of the uncertain graph in Figure 2() is 0.8. Remark 6 It is clear that the perfect matching index of the uncertain graph G in Example 2 is pκ(g) = 0. Example 3 Assume that G is an uncertain graph presented in Figure. Obviously, there are three spanning matchings in the uncertain graph G, and the matching index of the maximum index spanning matching is 0.9. According to Theorem 5, the matching index of the uncertain graph in Figure is

10 (): Uncertain graph G (4): Spanning matching P 3 4 (2): Spanning matching P (5): Spanning matching P (3): Spanning matching P (6): Spanning matching P 5 Figure 2: Spanning matchings for Example 2 4 Algorithm and Example In this section, we will give an algorithm to calculus the matching index, or say, find the maximum index spanning matching. Assume that uncertain graph G has vertex set V (G) = {v, v 2,, v n } and edge set E(G) = {ξ, ξ 2,, ξ m }. Suppose that P is a maximum matching of the underlying graph G. Algorithm Step. Sort {ξ, ξ 2,, ξ m } by M{ξ i = }. Set E = Φ, V = Φ and E = {ξ, ξ 2,, ξ m }. Step 2. Choose the edges with biggest M{ξ i = } in E, the set of these selected edges is denoted by A, the corresponding vertex set is U. Constructing a graph G with edge set E = E + A and vertex set V = V + U, and resetting E = E A. Step 3. Finding a maximum matching P in graph G. If P = P, then P is the maximum index spanning matching; Otherwise, go back to Step 2. In the algorithm, the Edmonds s matching algorithm (Edmonds [3]) can be used to find a maximum matching in Step 3. After the last iteration, Step 3 creates a spanning matching P such that P = P. Let P be a maximum index spanning matching. If P = P, we have done. Otherwise, there must exist a walk v 0 ξ 0 v ξ v 2 ξ 2t v 2t+ ξ 2t+ v 2t+2 in G such that E 0 = {ξ 0, ξ 2,, ξ 2t } P, E = {ξ, ξ 3,, ξ 2t+ } P and P = P + E 0 E is also a matching. What is more, the algorithm says that min M{ξ 2k = } min M{ξ 2k+ = }. According to Corollary, P is a spanning matching 0 k t 0 k t with κ(p ) κ(p ). Since P is the maximum index spanning matching, we have κ(p ) = κ(p ). That is, we obtain a maximum index spanning matching P that agrees more with P than P does. 0

11 Repeating this argument, we will finally obtain a maximum index spanning matching that agrees completely with P. Thus, the spanning matching obtained by the proposed algorithm is the maximum index spanning matching. Remark 7 When an uncertain graph G has a perfect matching index, then the designed algorithm is also can be used to calculus the perfect matching index. This fact follows that if G has a perfect matching index, then pκ(g) = κ(g). Figure 3 gives an example. According to Corollary, the matching index of the maximum index spanning matching in Figure 3(5) is 0.7, thus the matching index of the uncertain graph in Figure 3() is () (2) (3) (4) (5) Figure 3: Maximum index spanning matching by the Algorithm Remark 8 It is clear that the perfect matching index of the uncertain graph G in Figure 3() is equal to the matching index of G, i.e., pκ(g) = κ(g) = Conclusion An uncertain graph refers to the graph in which whether two vertices are joined by an edge cannot be completely determined. For a given uncertain graph, at how much belief degree we can regard the graph has a maximum matching? This paper mainly employed the uncertainty theory to investigate the matching problem in uncertain graph. The main contributions can be summarized as the following three aspects: () Matching index and perfect matching index were firstly proposed. (2) Some properties of matching index were investigated. (3) An algorithm was given to calculate the matching index of uncertain graph.

12 Sometimes, uncertainty and randomness will no doubt simultaneously appear in graphs. There is no doubt that the matching problem in such uncertain random environment may become a new topic in further research. Acknowledgements This work is supported by the National Natural Science Foundation (No ), the Hubei Provincial Natural Science Foundation (No.200CDB0280), and the Scientific and Technological Innovation Team Project (No.T200) of Hubei Provincial Department of Education, China. References [] B. Bollobás, Degree sequences of random graphs, Discret. Math. 33 (98) -9. [2] J. Bondy, U. Murty, Graph Theory with Applications, Elsevier, New York, 976. [3] J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 7 (965) [4] P. Erdös, A. Rényi, On random graph I, Publ. Math. Debr. 6 (959) [5] P. Erdös, A. Rényi, On the strength of connectedness of a random graph, Acta Math. Hungar. 2 (96) [6] X. Gao, Y. Gao, Connectedness index of uncertain graph, [7] Y. Gao, Shortest path problem with uncertain arc lengths, Comput. Math. Appl. 62 (20) [8] E. Gilbert, Random graphs, Ann. Math. Stat. 30 (959) [9] F. Harary, The maximum connectivity of a graph, Proc. Nat. Acad. Sci. USA 48 (962) [0] D. Kahneman, A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica 47 (979) [] X. Li, B. Liu, Hybrid logic and uncertain logic, J. Uncertain Syst. 6 (2009) [2] B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, [3] B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst. 2 (2008) 3-6. [4] B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 (2009) 3-0. [5] B. Liu, Theory and Practice of Uncertain Programming, 2nd ed., Springer-Verlag, Berlin, [6] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer- Verlag, Berlin, 200. [7] B. Liu, Uncertainty Theory, 4th ed., 2

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