THE inverse shortest path problem is one of the most

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1 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER An Inverse Shortest ath roblem on an Uncertain Graph Jian Zhou, Fan Yang, Ke Wang School of Management, Shanghai University, Shanghai , China Abstract The inverse shortest path problem is to minimize the modification on the edge weights such that a predetermined path becomes the shortest one from the origin to the destination with respect to the new edge weights. In this paper, the inverse shortest path problem is considered on a graph with uncertain edge weights. It is shown that the model of the uncertain inverse shortest path problem can be transformed into a deterministic counterpart and then be solved efficiently. A numerical eample is presented as well for illustration. Inde Terms Inverse shortest path problem, uncertainty theory, uncertain programming, network optimization I. INTRODUCTION THE inverse shortest path problem is one of the most typical problems of inverse optimization, which is to make a predetermined solution become the optimal solution after modifications. This problem has attracted many attentions recently due to its broad applications in practice such as the traffic modeling and the seismic tomography (see, e.g., [] [5]). In 992, Burton and Toint [6] first formulated the inverse shortest path problem using l 2 norm to measure the modification, which was solved by an algorithm for conve quadratic programming. Since then the inverse shortest path problem has been studied by many researchers. For instance, as a complement of Burton and Toint s work, Xu and Zhang [7] discussed the general structure of the feasible set of weight vectors in the inverse shortest path problem, and showed the relation between the inverse shortest path problem and the minimal cutset problem. Zhang et al. [8] formulated an inverse shortest path problem as a special linear programming problem under l norm, and developed a column generation algorithm that is likely more effective than the quadratic programming method to handle large size problems. Heuberger [9] remarked that the column generation algorithm has the advantage that no linear programming formulation of original problem has to be used, while the price is that the original problem has to be solved many times. Hu and Liu [0] designed an O(n 3 ) algorithm to solve the inverse shortest path problem using l norm. Corresponding Author: Ke Wang, ke@shu.edu.cn. This work was supported in part by grants from the Innovation rogram of Shanghai Municipal Education Commission (No. 3ZS065), the Shanghai hilosophy and Social Science lanning roject (No. 202BGL006), and the National Social Science Foundation of China (No. 3CGL057). Moreover, Zhang and Liu [] considered the cases in which the given feasible solution and optimal solution of the linear programming are 0- vectors which often occur in network programming and combinational optimization, and proved that the optimal solution of an inverse shortest path problem can be obtained from the dual problem under l norm. Farago et al. [5] applied the inverse shortest path problem to high speed telecommunication networks, combining the simplicity of fied routing and some advantages of a dynamic scheme to obtain reliable and stable self-configuring systems. The computational compleity of the inverse shortest path problem has also been investigated in some literature. For eample, Burton et al. [2] studied the computational compleity of the inverse shortest path problem with upper bounds on the sum of weights of the shortest path from the origin to the destination under l 2 norm, and proved that obtaining a globally optimum solution to this problem is N-complete. Similarly, Liu and He [3] proved that the inverse shortest path problem with discrete weights is strongly N-complete. In [4], Zhang et al. considered the shortest path improvement problem under hamming distance, which is a variant of the inverse shortest path problem, and discussed its computational compleity. Subsequently, Zhang et al. [5] designed two greedy algorithms for this problem on chain networks and special star-tree networks, one of which is a strongly polynomial time algorithm for the problem with a single source and constrained paths, and the other is a heuristic algorithm for general graphs. In most scenarios, the edge weights are assumed to be fied, but this does not always hold in practical applications. For eample, the traveling time may be affected by cars quantity, weather, road width and so on. In this case, some researchers believed that these nondeterministic phenomena conform to randomness or fuzziness, and hence the probability theory or the fuzzy set theory was introduced into the shortest path problem (see, e.g., [6] [9]). However, when no samples are available to estimate a probability distribution in the nondeterministic environment, we have to invite some domain eperts to evaluate the belief degree that each event will occur. erhaps some people think that the belief degree is subjective probability or fuzzy concept. However, it is usually inappropriate because both probability theory and fuzzy set theory may lead to counterintuitive results (see [20] for details). In order to deal with belief degrees rationally, the uncertainty theory was founded doi:0.4304/jnw

2 2354 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER 204 by Liu [20] in 2007 and subsequently applied to model the shortest path problem. For eample, Liu [2] defined three concepts of uncertain shortest path according to different decision criteria, involving uncertain epected shortest path, uncertain α-shortest path and uncertain most shortest path. Following that, Gao [22] gave the uncertainty distribution of the shortest path length, and pointed out that the uncertain α-shortest path on an uncertain graph is equivalent to the shortest path in a corresponding deterministic graph, and the equivalence leads to a stable and global optimal method to find the uncertain α-shortest path. Concerning the inverse shortest path problem, although much attention has been paid to the deterministic cases as mentioned above, it has not been adequately investigated in the uncertain situations yet. Therefore, this paper discusses the inverse shortest path problem on an uncertain graph based on the concept of uncertain α-shortest path proposed by Liu [2], which is formulated as an uncertain programming. We also show that when the edge weights are supposed to be independent uncertain variables with regular distributions, the uncertain inverse shortest path problem can be reformulated into a deterministic programming model according to the operational law of independent uncertain variables provided by Liu [23]. Furthermore, when the edge weights are assumed to be linear uncertain variables, the corresponding model reduces to a linear programming problem and can be solved efficiently taking the advantage of some well developed optimization software packages. The rest of the paper is organized as follows. Section II introduces the classical inverse shortest path problem, and some basic concepts of uncertain variables are reviewed. Section III defines the uncertain inverse shortest path problem and then formulates it as an uncertain programming. Section IV transforms the uncertain model into an equivalent deterministic programming. Section V presents a numerical eample for illustration. Finally, Section VI gives the conclusion of the whole paper. II. RELIMINARIES In this section, the classic inverse shortest path problem is briefly reviewed, and some basic notions and results of uncertain variables are recalled which are indispensable to handle the uncertain inverse shortest path problem. A. Classic Inverse Shortest ath roblem Generally speaking, the classic inverse shortest path problem is to assign some new weights to the edges of a graph such that a predetermined path is the shortest path from the origin to the destination with respect to new edge weights and the modification is minimized. In order to give the model of the inverse shortest path problem, let G = (V, E) be a connected graph consisting of verte set V = {v, v 2,, v n } and edge set E = {, 2,, m}. For each edge i E, there is an original positive weight c i as well as a new weight i after a modification. Besides, is the predetermined path from the origin to the destination. Then the inverse shortest path problem can be formulated as m min i c i () is the shortest path with respect to i > 0, i E. This model is to minimize the modification m i c i such that the predetermined path becomes the shortest path from the origin to the destination, where the term m i c i denotes the absolute deviation of the weights which is the most common type of the objective functions used in the inverse shortest path problem. Eample : An inverse shortest path problem with 6 vertices and 9 edges is shown in Fig. as a demonstration, in which c i and i denote the original and new weights of edge i, respectively. The edges (solid lines) define a path, i.e., AB-BC-CF, between the source A and the destination F. The objective of this problem is to make the path AB- BC-CF become the shortest one between A and F with minimum adjustments. A D (c 6, 6 ) (c 7, 7 ) B (c, ) (c 2, 2 ) (c 3, 3 ) (c 4, 4 ) E (c 5, 5 ) (c 8, 8 ) C (c 9, 9 ) Figure. An eample of the inverse shortest path problem B. Uncertain Variables Definition : (Liu [20]) Let L be a σ-algebra on a nonempty set Γ. A set function M : L [0, ] is called an uncertain measure if it satisfies the following aioms: Aiom : (Normality Aiom) M{Γ} = for the universal set Γ; Aiom 2: (Duality Aiom) M{Λ} + M{Λ c } = for any event Λ; Aiom 3: (Subadditivity Aiom) For every countable sequence of events Λ, Λ 2,, we have { } M Λ i M{Λ i }. In the uncertainty theory, the triplet (Γ, L, M) is called an uncertainty space, and the product uncertain measure on the product σ-algebra was defined by Liu [24] via the following product aiom: Aiom 4: (roduct Aiom) Let (Γ k, L k, M k ) be uncertainty spaces for k =, 2,. The product uncertain F

3 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } k= k= where Λ k are arbitrarily chosen events from L k for k =, 2,, respectively. An uncertain variable ξ is essentially a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable in practice, Liu [20] defined a concept of uncertainty distribution as follows. Definition 2: (Liu [20]) Let ξ be an uncertain variable. Then, its uncertainty distribution is defined by Φ() = M{ξ } (2) for any real number. Eample 2: An uncertain variable ξ is called linear if it has a linear uncertainty distribution (see Fig. 2) 0, if a Φ() = ( a)/(b a), if a b (3), if b denoted by L(a, b), where a and b are real numbers with a < b. Φ() 0 a b Figure 2. Linear uncertainty distribution An uncertainty distribution Φ is said to be regular if its inverse function (α) eists and is unique for each α (0, ). It is clear that a linear uncertain variable ξ L(a, b) has a regular distribution since its inverse uncertainty distribution (α) = ( α)a + αb (4) is unique for each α (0, ). The inverse uncertainty distribution plays an important role in the arithmetic operations of independent uncertain variables. Definition 3: (Liu [24]) The uncertain variables ξ, ξ 2,, ξ m are said to be independent if { m } m M (ξ i B i ) = M{ξ i B i } for any Borel sets B, B 2,, B m of real numbers. Theorem : (Liu [23]) Let ξ, ξ 2,, ξ m be independent uncertain variables with regular uncertainty distributions Φ, Φ 2,, Φ m, respectively. If the function f(, 2,, m ) is strictly increasing with respect to, 2,, k and strictly decreasing with respect to k+, k+2,, m, then ξ = f(ξ, ξ 2,, ξ m ) is an uncertain variable with inverse uncertainty distribution Ψ (α) = f( (α),, Φ k (α), k+ ( α),, Φ m ( α)). Eample 3: Let ξ be an uncertain variable with regular uncertainty distribution Φ. Since f() = a + b is a strictly increasing function for any constants a > 0 and b, it follows from Theorem that the inverse uncertain distribution of aξ + b is (5) Ψ (α) = a (α) + b. (6) rovided that ξ L(, 3), a = 3 and b = 0, we can get Ψ (α) = 6α + 3. It is easy to verify that aξ + b is a linear uncertain variable L(3, 9). Eample 4: Let ξ and ξ 2 be independent uncertain variables with regular uncertainty distributions Φ and Φ 2, respectively. It follows from Theorem that the inverse uncertainty distribution of the difference ξ ξ 2 is Ψ (α) = (α) Φ 2 ( α). (7) rovided that ξ L(4, 0) and ξ 2 L(2, 5), we have Ψ (α) = 9α. It is easy to verify that ξ ξ 2 is a linear uncertain variable L(, 8). In fact, for independent uncertain variables with linear uncertainty distributions, they possess some good properties under additive and scalar multiplication. That is, if ξ L(a, b ), ξ 2 L(a 2, b 2 ), and ξ, ξ 2 are independent, then for k > 0 and k 2 > 0, k ξ + k 2 ξ 2 L(k a + k 2 a 2, k b + k 2 b 2 ). (8) Moreover, it can be verified that for k > 0 and k 2 > 0, k ξ +k 2 ξ 2 (α) = k ξ (α) + k 2 ξ 2 (α) (9) where ξ and ξ 2 are independent uncertain variables with regular uncertainty distributions. It follows immediately from Theorem. III. UNCERTAIN INVERSE SHORTEST ATH ROBLEM Let G = (V, E) be a connected graph consisting of verte set V = {v, v 2,, v n } and edge set E = {, 2,, m}. For each edge i E, there is a weight ξ i, which is assumed to be an uncertain variable with regular distribution Φ i. Suppose that is the predetermined path from the origin to the destination. For simplicity, we denote a path by its edge set throughout this paper. Then the length of a path is defined as T (ξ) = i ξ i (0) which is also an uncertain variable, and the uncertainty distribution of the path length T is denoted by Ψ.

4 2356 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER 204 In order to formulate the inverse shortest path problem on an uncertain graph, let us introduce the notion of uncertain α-shortest path as follows. Definition 4: (Liu [2]) A path is called the uncertain α-shortest path from the origin to the destination if min{t M{T 0 (ξ) T } α} () min{t M{T (ξ) T } α} for all paths from the origin to the destination, where α is a predetermined confidence level. Definition 4 is based on a given confidence level α, which means the chance of achieving the goal. It should be noted that () can be rewritten as Ψ (α) Ψ (α) (2) where Ψ and Ψ are the inverse uncertainty distributions of path lengths T 0 and T, respectively (see Fig. 3). α α 0 0 Ψ (α 0 )Ψ (α 0) Ψ 0 Figure 3. Uncertain α-shortest path Ψ Now let us define the inverse shortest path problem on an uncertain graph with uncertain edge weights ξ i, i E. For this purpose, we assume that there is a parameter c i on each edge i E, which will be modified to i after adjustments. Moreover, the edge weights ξ i are assumed to be related to this parameter. In other words, for each edge i E, there is an original weight ξ i (c i ) as well as a new weight ξ i ( i ). Given a path from the origin to the destination, the uncertain inverse shortest path problem is to find new parameters i such that becomes an uncertain α-shortest path with respect to the new edge weights ξ i ( i ), i E, and the total change of the parameters, i.e., m i c i, is minimized. Eample 5: The following traffic network improvement problem is a typical uncertain inverse shortest path problem. A traffic network consisting of 6 vertices and 9 edges is shown in Fig. 4, where the vertices denote communities in a town, and the edges denote the roads between the communities. The travelling time ξ i between two communities is an uncertain variable due to the comple real situations, such as weather and traffic jam. It is also related to the resources assigned to this road, e.g., the number of buses serving on this road. Denoting by c i T the initially assigned resources, the original travelling time can be formulated as ξ i (c i ). Now, the traffic management office wants to modify or redesign the traffic network by adjusting the resources from c i to i according to the development of the town. The predetermined path denoted by solid lines, i.e., AB-BC-CF, is required to be the shortest one with respect to the new travelling time ξ i ( i ) for some political or economic reasons. In order to decrease the cost of adjustments as possible, the modification is to be minimized. A D (c 6, 6,ξ 6 ) (c 7, 7,ξ 7 ) B (c,,ξ ) (c 2, 2,ξ 2 ) (c 3, 3,ξ 3 ) (c 4, 4,ξ 4 ) E (c 5, 5,ξ 5 ) (c 8, 8,ξ 8 ) C (c 9, 9,ξ 9 ) Figure 4. An eample of the uncertain inverse shortest path problem Based on Definition 4 and (2), the uncertain inverse shortest path problem can be formulated as an uncertain programming as follows, m min i c i (3) Ψ (; α) Ψ (; α), Ω i > 0, i E where α is a predetermined confidence level, Ω is the set of paths from the origin to the destination, Ψ and Ψ are the inverse uncertainty distributions of path lengths T 0 and T, respectively. This model means to use the least adjustments of parameters to make the predetermined path become an uncertain α-shortest path. IV. CRIS EQUIVALENT MODEL In this section, the uncertain programming (3) is transformed to a crisp equivalent model based on the operational law of independent uncertain variables (see Theorem ). Theorem 2: Suppose that the edge weights ξ i are independent uncertain variables with regular distributions Φ i, i E, respectively. Then for any path Ω, the inverse uncertainty distribution of the path length T (ξ) is Ψ (α) = i (α) (4) i where i are the inverse uncertainty distributions of uncertain weights ξ i, i E, respectively. roof: It follows immediately from Theorem. F

5 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER By Theorem 2, model (3) can be reformulated as m min i c i i ( i ; α) (5) i ( i ; α), Ω i i i > 0, i E. Note that for each i E, the inverse distribution i is related to the new parameter i. Once uncertainty distributions are specified, model (5) becomes a deterministic programming problem and may be solved by some well developed algorithms or software packages. Generally, it is a nonlinear programming problem and may require much computational effort. However, it is easy to see that model (5) is nothing but a deterministic inverse shortest path problem. In other words, the uncertain inverse shortest path problem can be handled within the framework of the classic deterministic inverse shortest path problem requiring no particular solving methods in such an uncertain environment. When the uncertain edge weights ξ i, i E, are linear uncertain variables, for eample, ξ i L( i d i, i +d i ), it follows from (4) that the inverse uncertainty distribution of ξ i is i ( i ; α) = ( α)( i d i ) + α( i + d i ) = i + d i (2α ). (6) Consequently, model (5) reduces to a deterministic programming problem with linear constraints as follows, m min i c i ( i + d i (2α )) (7) i ( i + d i (2α )), Ω i i > 0, i E. Furthermore, by introducing auiliary variables + i, i, i E, where { + i c i, if i c i i = 0, if i c i, i = { 0, if i c i c i i, if i c i, the terms i c i, i E, can be represented as By (8), we have (8) i c i = + i + i. (9) i = + i i + c i, + i 0, 0 i < c i. (20) Taking (9) and (20) into model (7), a model can be obtained as follows, m m min + i + i ( + i i + c i + d i (2α )) i ( + i i + c i + d i (2α )), Ω i + i 0, i E 0 i < c i, i E (2) which is eactly a deterministic linear programming model. Until now, it is shown that if the uncertain edges ξ i, i E, are assumed to be independent linear uncertain variables with regular uncertainty distributions, the uncertain inverse shortest path problem can be transformed to a linear programming problem, and many effective methods for linear programming can be applied to solving it. V. NUMERICAL EXAMLE A numerical eample of uncertain inverse shortest path problem with 6 vertices and 0 edges is shown in Fig. 5, where the edges denoted by solid lines define a predetermined path between sites A and F, i.e., = {, 9, 0}. For each edge i E = {, 2,, 0}, the weight ξ i is a linear uncertain variable with distribution L( i 0, i + 0), where i is the decision variable representing a parameter with an original value c i. The values of c i and ξ i, i E, are listed in Table I. (c,,ξ ) A (c 6, 6,ξ 6 ) B D (c 3, 3,ξ 3 ) (c 9, 9,ξ 9 ) (c 8, 8,ξ 8 ) C (c 2, 2,ξ 2 ) (c 4, 4,ξ 4 ) (c 7, 7,ξ 7 ) E (c 5, 5,ξ 5 ) F (c 0, 0,ξ 0 ) Figure 5. Uncertain graph for the numerical eample According to model (3), if we want to minimize the total modification so as to diminish the total cost of adjustments with a confidence level α = 0.8, we have the following uncertain programming, 0 min i c i (22) Ψ (; 0.8) Ψ (; 0.8), Ω i > 0, i =, 2,, 0

6 2358 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER 204 TABLE I. ARAMETER VALUES IN FIG. 5 Edge Original arameter Uncertain Edge Weight i c i ξ i ( i ) 30 L( 0, + 0) 2 50 L( 2 0, 2 + 0) 3 40 L( 3 0, 3 + 0) 4 80 L( 4 0, 4 + 0) 5 50 L( 5 0, 5 + 0) 6 40 L( 6 0, 6 + 0) 7 90 L( 7 0, 7 + 0) 8 60 L( 8 0, 8 + 0) 9 70 L( 9 0, 9 + 0) 0 40 L( 0 0, 0 + 0) settings and results including the optimal solutions and the minimum objective values are summarized in Table II. TABLE II. RESULTS FOR THE NUMERICAL EXAMLE USING DIFFERENT CONFIDENCE LEVELS α OBJ where Ω is the set of all the paths between sites A and F in Fig. 5, Ψ and Ψ are the inverse uncertainty distributions of the predetermined path and any path between A and F, i.e., Ω. By Theorem 2, we can transform the above model into a deterministic programming as follows, 0 min i c i i ( i ; 0.8) i ( i ; 0.8), Ω i 0 i i > 0, i =, 2,, 0 (23) where i are the inverse uncertainty distributions of ξ i L( i 0, i + 0), i =, 2,, 0. It follows from (6) that i ( i ; 0.8) = i + 0(2 0.8 ) = i + 6. (24) Consequently, by defining + i and i as formulated in (8), and then taking (9), (20) and (24) into model (23), it becomes 0 0 min + i + i ( + i i + c i + 6) i ( + i i + c i + 6), Ω i + i 0, i =, 2,, 0 0 i < c i, i =, 2,, 0. (25) Using MATLAB optimization package, an optimal solution is found at = (30, 27, 48, 80, 57, 40, 30, 60, 70, 36) with an objective value 02. The predetermined confidence level α plays an important role in the formulation. In order to investigate the influence of this parameter, we further consider the numerical eample for different confidence levels. The As shown in Table II, the objective value increases when a higher confidence level is taken. In other words, the higher confidence level the decision-maker demands, the more cost is needed. VI. CONCLUSION In this paper the inverse shortest path problem is discussed in the uncertain environment, where the edge weights are assumed to be independent uncertain variables with regular distributions. Based on the uncertain α- shortest path defined by Liu [2], the uncertain inverse shortest path problem is formulated as an uncertain programming, and then transformed into a deterministic model according to the operational law of independent uncertain variables. It is shown that the uncertain inverse shortest path problem can be handled within the framework of the classic deterministic inverse shortest path problem requiring no particular solving methods. When the edge weights are linear uncertain variables, this model further reduces to a linear programming and hence can be solved efficiently with the aid of some well developed optimization software packages. ACKNOWLEDGMENT The authors are grateful to the anonymous referees for their valuable comments and suggestions to improve the presentation of this paper. REFERENCES [] G. Neumann-Denzau and J. Behrens, Inversion of seismic data using tomographical reconstruction techniques for investigations of laterally inhomogeneous media, Geophysical Journal International, vol. 79, no., pp , 984. [2] G. Nolet, Seismic Tomography, Reidel, Dordrech, 987. [3] D. C. Wei, An optimized floyd algorithm for the shortest path problem, Journal of Networks, vol. 5, no. 2, pp , 200. [4] Q. S. Wu, C. Y. Tong, Q. Wang, and X. F. Cheng, All-pairs shortest path algorithm based on MI+CUDA distributed parallel programming model, Journal of Networks, vol. 8, no. 2, pp , 203. [5] A. Faragó, A. Szentesi, and B. Szviatovszki, Inverse optimization in high-speed networks, Discrete Applied Mathematics, vol. 29, no., pp , 2003.

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