The α-maximum Flow Model with Uncertain Capacities
|
|
- Winifred Wiggins
- 6 years ago
- Views:
Transcription
1 International April 25, 2013 Journal7:12 of Uncertainty, WSPC/INSTRUCTION Fuzziness and Knowledge-Based FILE Uncertain*-maximum*Flow*Model Systems c World Scientific Publishing Company The α-maximum Flow Model with Uncertain Capacities Sibo Ding Uncertainty Operations Research Laboratory, School of Management Henan University of Technology, Zhengzhou , China dingsibo@haut.edu.cn Uncertain theory is a new tool to deal with the maximum flow problem with uncertain arc capacities. This paper investigates uncertain maximum flow problem and presents the uncertainty distribution of the maximum flow. Uncertain α-maximum flow model is formulated. It is proved that there exists an equivalence relationship between uncertain α-maximum flow model and the classic deterministic maximum flow model, which builds a bridge between uncertain maximum flow problem and deterministic maximum flow problem. Furthermore, some important properties of the model are analyzed, based on which a polynomial exact algorithm is proposed. Finally, a numerical example is presented to illustrate the model and the algorithm. Keywords: Maximum flow problem; Uncertainty theory; Uncertain Programming; Generic preflow-push algorithm. 1. Introduction The maximum flow problem is one of the core issues of network optimization and has been widely studied. This problem was first investigated by Fulkerson and Dantzig 1. Then, Ford and Fulkerson 2 solved it using augmenting path algorithm. Motivated by a desire to develop a method with improved worst-case complexity, Dinic 3 introduced the concept of layered networks. His algorithm proceeds by augmenting flows along directed paths from source to sink in the layered network. Edmonds and Karp 4 also independently proposed that the Ford and Fulkerson algorithm augments flow along shortest paths. Until this point all maximum flow algorithms were augmenting path algorithms. However, the augmenting path algorithm could be slow because it might perform a large number of augmentations. In order to reduce the number of augmentations, Karzanov 5 introduced the first preflow-push algorithm on layered networks. Goldberg and Tarjan 6 constructed distance labels instead of layered networks to improve the running time of preflow-push algorithm. They described a very flexible generic preflow-push algorithm that performs push and relabel operations at active nodes. Their algorithm can examine active nodes in any order. In practice, flow capacities may change over time in communication network or transportation network, and one may assume network arcs have different values (capacity, cost) that are random variables with known probability distributions. As an extension of deterministic maximum flow problem, the stochastic maximum flow problem has been investigated extensively. Frank and Hakimi 7 assumed that each branch Crresponding author 1
2 2 Sibo ding in communication network has a random capacity and attempted to find the probability of a flow between vertices. Frank and Frisch 8 considered how to determine the maximum flow probability distribution in networks where each capacity is a continuous random variable. Furthermore, Doulliez 9 studied multiterminal network with discrete probabilistic branch capacities. In addition, some researchers have tried to give lower and upper bounds on the expected maximum flow. Onaga 10 derived an upper bound in general undirected or directed networks, while Carey and Hendrickson 11 presented a efficient method to find a lower bound in general directed networks. After that, Nagamochi and Ibaraki 12 provided necessary and sufficient conditions for Carey and Hendrickson s lower bound. In reality, however, there exists indeterminacy about the parameters (capacities, costs) of maximum flow problems. That indeterminacy cannot be described by random variable because no samples are available. For instance, when networks subject to extreme events such as earthquakes, it is impossible to get probability distribution of arc capacities. If we insist on using probability theory to deal with indeterminacy, counterintuitive results will occur 13. But experts can estimate, based on their experience, the belief degree that arc capacities are less than or equal to a given value. In order to deal with this kind of human uncertainty, Liu 14 founded uncertainty theory and refined it 15. Since then, uncertainty theory and its application have experienced explosive growth. Peng and Iwamura 16 derived sufficient and necessary condition for uncertainty distribution. Liu and Ha 17 developed a formula for calculating the expected values of monotone functions of uncertain variables. In order to deal with mathematical programming involving with uncertain parameters, Liu 18 first proposed uncertain programming theory to model uncertain optimization problems. After that, Liu and Yao 19 introduced an uncertain multilevel programming for modeling uncertain decentralized decision systems, and Liu and Chen 20 developed an uncertain multiobjective programming and an uncertain goal programming. Some attempts have been made to collect expert s experimental data and get uncertain distribution. Liu 21 suggested the principle of least squares to estimate the unknown parameters of uncertainty distribution. Moreover, when a number of experts are available, Wang, Gao and Guo 22 applied the Delphi method to determine the uncertainty distributions. Chen and Ralescu 23 used B-spline method to estimate the uncertainty distribution. Other particular care has been taken to investigate uncertain graph and uncertain network. The connectedness index of uncertain graph was proposed by Gao and Gao 24. Zhang and Peng 25 suggested a method to calculate Euler index of uncertain graph. Gao 26 computed cycle index of uncertain graph. As an important contribution, Liu 21 first introduced uncertainty theory into network optimization. He studied project scheduling problem with uncertain duration times. Furthermore, Liu 27 assumed uncertainty and randomness simultaneously appear in a complex network, and put forward the concept of uncertain random network. Gao 28 gave an equivalence relation between the uncertain α-shortest path and the deterministic shortest path. And, Han and Peng 29 studied the uncertain maximum flow problem and provided a numerical solution method. This paper constructs an uncertain α-maximum flow model, analyzes properties of the model, and designs a new polynomial-time exact algorithm. The remainder of this paper is organized as follows. In section 2, some basic concepts and properties of uncertainty theory used throughout this paper are introduced. In section 3, Uncertain α-maximum flow model is formulated and its properties are analyzed. In section 4, an optimal algorithm is developed to solve the model. A numerical example is presented to illustrate the algorithm in Section 5. Section 6 gives a conclusion to this paper.
3 Uncertain α-maximum Flow Model 3 2. Preliminaries Uncertainty theory is a branch of mathematics for modeling human uncertainty. In order to deal with human uncertainty, Liu 14,15 presented four axioms: (1) normality axiom, (2) duality axiom, (3) subadditivity axiom, and (4) product axiom. In this section, we introduce some fundamental concepts and properties of uncertainty theory, which will be used throughout this paper. Definition Let Γ be a nonempty set, L a σ-algebra over Γ, and M an uncertain measure. Then the triplet (Γ, L, M) is called an uncertainty space. Definition An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} = {γ Γ ξ(γ) B} is an event. Definition The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x) = M{ξ < x} for any real number x. The zigzag uncertainty distribution ξ Z(a, b, c) has an uncertainty distribution 0, if x a (x a)/2(b a), if a x b Φ(x) = (x + c 2b)/2(c b), if b x c 1, if x c. Definition An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < M(x) < 1, and lim x Φ(x) = 0, lim x + Φ(x) = 1. Definition The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if { n } n M (ξ i B i ) = M{ξ i B i } i=1 for any Borel sets B 1, B 2,, B n of real numbers. i=1 Theorem Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n )
4 4 Sibo ding is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). In reality, we can easily obtain strictly increasing functions f. Therefore, through Theorem 1, we can transform an indeterminacy model into a deterministic one by to its equivalent deterministic form M{f(ξ 1, ξ 2,, ξ n ) x} α, x f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). 3. Mathematical formulation This paper concerns uncertain maximum flow problem. The goal is to maximize the total flow sent from the source node to the sink node not exceeding the capacities on any arc and keeping flow balance in every node. In classic deterministic maximum flow problem, a flow network is 5-tuple N = (V, A, c, s, t), with source node s, sink node t, a finite node set V = {1, 2,, n} and arc set A = {(i, j) i, j V }, together with a nonnegative real-valued capacity function c defined on its arc set A. Denote u = {u ij (i, j) A} as the set of arc capacities. Then in network N, from the source s to the sink t, the maximum flow is a function of u, which is denoted as f. Given u, f(u) can be found. Denote x = {x ij (i, j) A} as the set of flow on arc (i, j). A flow is feasible if it satisfies conservation condition x sj x js = f, (i, j) A j:(s,j) A j:(j,s) A x ij x ji = 0, (i, j) A j:(i,j) A j:(t,j) A x tj j:(j,i) A j:(j,t) A x jt = f, (i, j) A and capacity constraint 0 x ij u ij, where f is the flow in the network N. In classic deterministic maximum flow problem, capacities of arcs are scrip values. Unfortunately, cases such as this which capacities of arcs are scrip values are rare. Especially, networks are quite large, and it is impossible to describe them explicitly. But, if there are enough data available, random maximum flow models may be considered as network models. However, in most cases, we cannot get enough data or data is invalid because of change in conditions. For example, due to impact of unexpected accidents on traffic flow, we cannot use probabilistic rules to describe the complexity of the networks. In this situation, the capacity data can only be obtained from the decision-maker s subjective estimation. Thus, it is unsuitable to regard subjective estimation data as random variables. In this paper, we employ uncertain variables to describe the capacities of arcs. We consider the maximum flow problem subject to the following assumptions: (1) The network is directed. (2) All capacities are nonnegative rational numbers (All computers store capacities as rational numbers and we can always transform rational numbers to integer numbers by multiplying them by a suitably large number). (3) The network does not contain a directed path from source node s to sink node t composed only of infinite capacity arcs. (4) The network does not contain parallel arcs.
5 Uncertain α-maximum Flow Model 5 Define ξ = {ξ ij (i, j) A}. We can denote the network with uncertain capacities of arcs as N = (V, A, ξ, s, t). The maximum flow is f(ξ). As a function of ξ, f is also an uncertain variable. Sometimes, the decision-maker assumes that the flow should satisfy some chance constraints with at least some given confidence level α. Then we have the following definition. Definition 1. A flow x is the α-maximum flow from source s to sink t if max{f M{ξ x} α} max{f M{ξ x } α} for any flow x from source s to sink t, where α is a predetermined confidence level. Chance constrained programming offers a powerful tool for modeling uncertain decision systems. The essential idea of chance constrained programming of α maximum flow model is to optimize the flow value in network with predetermined confidence level subject to capacity chance constraints. In order to find α-maximum flow, we propose the following uncertain α-maximum flow model. max f subject to : x ij f, i = s, x ji = 0, i V {s, t}, j:(i,j) A j:(j,i) A f, i = t, M{ξ ij x ij } α, (i, j) A x ij 0, (i, j) A, (1) where α is a predetermined confidence level provided by the decision-maker. In classic deterministic model, the maximum flow is obtained by polynomial algorithms. It implies that if there is one way to transform uncertain α-maximum model into its crisp equivalent, then we can solve the model in deterministic environment by polynomial algorithms. Therefore, we need to convert the chance constraints M{ξ ij x ij } α, (i, j) A into its crisp equivalent. In order to deign algorithm for α-maximum flow model, we first introduce property of the chance constraint. Lemma 1. According to Theorem 1, M{ξ ij x ij } α, for (i, j) A can be transformed as x ij Φ 1 ij (α). Proof. Denote the uncertainty distribution of ξ ij as Φ ij, for (i, j) A. According to Theorem 1, for any 0 < α < 1, M{ξ ij Φ 1 ij (α)} = α. Since uncertainty distribution is an increasing function, from M{ξ ij x ij } α = M{ξ ij Φ 1 ij (α)},
6 6 Sibo ding we obtain This proves the lemma. x ij Φ 1 ij (α). Lemma 1 indicates that the chance constraint can be converted into its crisp equivalent, and we can draw the following theorem. Theorem 2. In network N = (V, A, ξ, s, t), ξ ij has a regular uncertainty distribution Φ ij, (i, j) A. Then, α-maximum flow of N = (V, A, ξ, s, t) is just the maximum flow of the corresponding deterministic network Ñ = (V, A, c, s, t), where the capacity of arc (i, j) A is Φ 1 ij (α). Proof. By Lemma 1, model (1) can be easily converted into the following deterministic model: max f subject to : x ij f, i = s, x ji = 0, i V {s, t}, j:(i,j) A j:(j,i) A f, i = t, (2) x ij Φ 1 ij (α), (i, j) A x ij 0, (i, j) A Thus, the solution to model (2) is just the maximum flow of deterministic network Ñ = (V, A, c, s, t), where the capacity of arc(i, j) A is Φ 1 ij (α). Thus, We can obtain the maximum flow of Ñ = (V, A, c, s, t) by using generic preflow-push algorithm. The theorem is proved. Theorem 1 shows how to obtain α-maximum flow. Next, we further investigate the property of the model for obtaining the inverse distribution of uncertain maximum flow. Theorem 3. In network N = (V, A, ξ, s, t), ξ ij has a regular uncertainty distribution Φ ij. Then, the inverse uncertainty distribution of f is determined by Ψ 1 (α) = f(φ 1 ij (α) (i, j) A). Proof. For model (1), the maximum flow f is a continnuous and increasing function with respect to each capacity of arc. Obviously, increasing the capacity of each arc, we will get a greater flow. That is, f(x) > f(y), where x = {x ij (i, j) A}, y = {y ij (i, j) A}, and x ij > y ij. Thus, f is a strictly increasing function. By Theorem 1 and Theorem 2, we can easily obtain the inverse uncertainty distribution of f with respect to α. The theorem is proved.
7 Uncertain α-maximum Flow Model 7 4. Solution algorithm Generally, Monte Carlo simulation or heuristic algorithms are used to obtain solution to uncertain programming with indeterminacy factors. A disadvantage of these methods in comparison to exact mathematical methods is that they usually provide only statistical estimates or approximate solutions, not exact result. Theorem 2 provides a better way to obtain the α-maximum flow in uncertain network. We only need to employ the generic preflow-push algorithm to find the maximum flow of the corresponding deterministic network. Hence, based on Lemma 1 and Theorem 2, we can design the following optimal solution algorithm for obtaining α-maximum flow. Algorithm: Step 1. Give a predetermined confidence level α and calculate Φ 1 ij (α), (i, j) A. Step 2. Construct the corresponding deterministic network Ñ = (V, A, c, s, t), and set the capacity of each arc u ij equal to Φ 1 ij (α). Step 3. Empoly the generic preflow-push algorithm to find α-maximum flow in network N. This algorithm runs in O(n 2 m) time (n is the number of nodes and m is the number of arcs) which is the same as that of generic preflow-push algorithm. 5. Numerical example In this section, we give an example to illustrate the algorithm. A network N = (V, A, ξ, s, t) is shown in Fig.1. The α-maximum flow is f with distribution Ψ. The capacity of each arc (i, j) is listed in Table 1. For convenience, if ξ ij is a constant, we set Φ 1 ij (α) = c, for any α (0, 1). Thus, we can calculate Φ 1 ij (0.9) for each ξ ij. Values of Φ 1 ij (0.9) are also listed Table 1. Then, we can obtain the following results: (1) the α-maximum flow when α = 0.9; (2) the uncertainty distribution of f(ξ). Fig. 1. Network for example.
8 8 Sibo ding Table 1. List of arc capacities and Φ 1 ij (0.9). arc (i, j) ξ ij Φ 1 ij (0.9) (s, 1) Z(14, 16, 18) 17.6 (1, 2) 6 6 (1, 3) Z(9, 10, 11) 10.8 (2, t) Z(10, 11, 12) 11.8 (3, t) Z(10, 12, 14) 13.6 Fig.2 Using the data in Table 1, we construct the deterministic network Ñ = (V, A, c, s, t). It is shown in Fig. 2. Network with upper bounds. Then, we solve α-maximum flow problem using the generic preflow-push algorithm. We obtain the final residual network given in Fig.3 (e( ) are excesses of nodes, d( ) are distance labels and r ij is residual capacity of any arc (i, j) A ). Fig. 3. Final residual network. Now the network contains no active node. The α-maximum flow in the network is shown in Fig.4 and its value is Choosing different α and repeating the above process, we obtain the uncertainty distribution of f, which is listed in Table 2 and plotted in Fig.5.
9 Uncertain α-maximum Flow Model 9 Fig. 4. The optimal solution to example. Table 2. List of α-maximum flows. α x s1 x 12 x 13 x 2t x 3t f = Ψ 1 (α) α f Fig. 5. Uncertainty distribution of f. 6. Conculsion Interpretations of indeterminacy phenomena may vary from time to time or person to person. Uncertainty theory provides a new tool to deal with indeterminacy. Under the framework of uncertainty theory, we present an extension to the classic maximum flow problem whose network capacities are uncertain variables in stead of crisp values. The problem was formulated by uncertain α-maximum flow model.
10 10 Sibo ding The relationship between the uncertain α-maximum flow model and its crisp equivalent is proved, and the uncertainty distribution of the uncertain maximum flow is derived. Some important properties of the model are analyzed, which help to develop a polynomial time exact algorithm. At last, an uncertain α-maximum flow example is given and solved by the algorithm. 7. Acknowledgments This work was supported by the National Natural Science Foundation of China Grant No and Education Research Project No.YPGC2011-W03. References 1. D. R. Fulkerson and G. B. Dantzig, Computations of maximum flow in networks, Naval Res. Log. Quart. 2 (4) (1955) G. B. Dantzig and D. R. Fulkerson, On the max-flow min-cut theorem of networks, in Linear Inequalities and Related Systems, eds. H. W. Kuhn and A. W. Tucker (Princeton University Press, Princeton, 1956) pp E. A. Dinic, Algorithm for solution of a problem of maximum flow in networks with power estimation,soviet Math. Dokl. 11 (1970) J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM 19 (2) (1972) A. V. Karzanov, Determining the maximal flow in a network by the method of preflows, Soviet Math. Dokl. 15 (2) (1974) A. V. Goldberg and R. E. Tarjan, A new approach to the maximum flow problem, Proc. Of the 18th Annual ACM Symposium on the Theory of Computing, Berkeley, CA, May, 1986, pp H. Frank and S. L. Hakimi, Probabilistic flows through a communication network, IEEE Trans. on Circuit Theory 12 (3) (1965) H. Frank and I. T. Frisch, Communication, Transmission, and Transportation Networks (Addison-Wesley, Reading, 1971). 9. P. Doulliez, Probability distribution function for the capacity of a multiterminal network, Rev. Franc. Inform. Rech. Oper. 5 (V1) (1971) K. Onaga, Bounds on the average terminal capacity of probabilistic nets, IEEE Trans. Inf. Theory 14 (5) (1968) M. Carey and C. Henrickson, Bounds on expected performance of networks with links subject to failure, Networks 14 (3) (1986)
11 Uncertain α-maximum Flow Model H. Nagamochi and T. Ibaraki, Maximum flows in probabilistic networks, Networks 21 (6) (1991) B. Liu, Why is there a need for uncertainty theory?, J. Uncertainty Syst. 6 (1) (2012) B. Liu, Uncertainty Theory, 2nd ed. (Springer-Verlag, Berlin, 2007). 15. B. Liu, Some research problems in uncertainty theory, J. Uncertainty Syst. 3 (1) (2009) Z.X. Peng and K. Iwamura, A sufficient and necessary condition of uncertainty distribution, Information 13 (3) (2010) Y.H. Liu and M.H. Ha, Expected value of function of uncertain variables, J. Uncertainty Syst. 4 (3) (2010) B. Liu, Theory and Practice of Uncertain Programming, 2nd ed. (Springer-Verlag, Berlin, 2009). 19. B. Liu and K. Yao, Uncertain multilevel programming: Algorithm and applications, B. Liu and X.W Chen, Uncertain multiobjective programming and uncertain goal programming (Technical Report, 2013). 21. B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty (Springer-Verlag, Berlin, 2010). 22. X.S. Wang, Z.C. Gao and H.Y. Guo, Uncertain hypothesis testing for two experts empirical data Math. Comput. Model. 55 (3-4) (2012) X.W. Chen and D.A. Ralescu, B-spline method of uncertain statistics with applications to estimate travel distance, J. Uncertainty Syst. 6 (4) (2012) L.X. Gao and Y. Gao, Connectedness index of uncertain graphs, Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 21 (1) (2013) B. Zhang and J. Peng, Euler index in uncertain graph, Appl. Math. Comput. 218 (20) (2011) X.L. Gao, Cycle index of uncertain graph, Information 16 (2A) (2013) B. Liu, Uncertain random graphs and uncertain random networks (Technical Report, 2013). 28. Y. Gao, Shortest path problem with uncertain arc lengths, Comput. Math. Appl. 62 (6) (2011) S.W. Han and Z.X. Peng, The maximum flow problem of uncertain network,
Minimum Spanning Tree with Uncertain Random Weights
Minimum Spanning Tree with Uncertain Random Weights Yuhong Sheng 1, Gang Shi 2 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China College of Mathematical and System Sciences,
More informationMembership Function of a Special Conditional Uncertain Set
Membership Function of a Special Conditional Uncertain Set Kai Yao School of Management, University of Chinese Academy of Sciences, Beijing 100190, China yaokai@ucas.ac.cn Abstract Uncertain set is a set-valued
More informationGraphs and Network Flows IE411. Lecture 12. Dr. Ted Ralphs
Graphs and Network Flows IE411 Lecture 12 Dr. Ted Ralphs IE411 Lecture 12 1 References for Today s Lecture Required reading Sections 21.1 21.2 References AMO Chapter 6 CLRS Sections 26.1 26.2 IE411 Lecture
More informationSpanning Tree Problem of Uncertain Network
Spanning Tree Problem of Uncertain Network Jin Peng Institute of Uncertain Systems Huanggang Normal University Hubei 438000, China Email: pengjin01@tsinghuaorgcn Shengguo Li College of Mathematics & Computer
More informationA New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle
INFORMATION Volume xx, Number xx, pp.54-63 ISSN 1343-45 c 21x International Information Institute A New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle Zutong Wang 1, Jiansheng
More informationMatching Index of Uncertain Graph: Concept and Algorithm
Matching Index of Uncertain Graph: Concept and Algorithm Bo Zhang, Jin Peng 2, School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2 Institute of Uncertain Systems, Huanggang
More informationUncertain Risk Analysis and Uncertain Reliability Analysis
Journal of Uncertain Systems Vol.4, No.3, pp.63-70, 200 Online at: www.jus.org.uk Uncertain Risk Analysis and Uncertain Reliability Analysis Baoding Liu Uncertainty Theory Laboratory Department of Mathematical
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationMinimum spanning tree problem of uncertain random network
DOI 1.17/s1845-14-115-3 Minimum spanning tree problem of uncertain random network Yuhong Sheng Zhongfeng Qin Gang Shi Received: 29 October 214 / Accepted: 29 November 214 Springer Science+Business Media
More informationKnapsack Problem with Uncertain Weights and Values
Noname manuscript No. (will be inserted by the editor) Knapsack Problem with Uncertain Weights and Values Jin Peng Bo Zhang Received: date / Accepted: date Abstract In this paper, the knapsack problem
More informationVariance and Pseudo-Variance of Complex Uncertain Random Variables
Variance and Pseudo-Variance of Complex Uncertain andom Variables ong Gao 1, Hamed Ahmadzade, Habib Naderi 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China gaor14@mails.tsinghua.edu.cn.
More informationInclusion Relationship of Uncertain Sets
Yao Journal of Uncertainty Analysis Applications (2015) 3:13 DOI 10.1186/s40467-015-0037-5 RESEARCH Open Access Inclusion Relationship of Uncertain Sets Kai Yao Correspondence: yaokai@ucas.ac.cn School
More informationEuler Index in Uncertain Graph
Euler Index in Uncertain Graph Bo Zhang 1, Jin Peng 2, 1 School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2 Institute of Uncertain Systems, Huanggang Normal University
More informationTail Value-at-Risk in Uncertain Random Environment
Noname manuscript No. (will be inserted by the editor) Tail Value-at-Risk in Uncertain Random Environment Yuhan Liu Dan A. Ralescu Chen Xiao Waichon Lio Abstract Chance theory is a rational tool to be
More informationHamilton Index and Its Algorithm of Uncertain Graph
Hamilton Index and Its Algorithm of Uncertain Graph Bo Zhang 1 Jin Peng 1 School of Mathematics and Statistics Huazhong Normal University Hubei 430079 China Institute of Uncertain Systems Huanggang Normal
More informationReliability Analysis in Uncertain Random System
Reliability Analysis in Uncertain Random System Meilin Wen a,b, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b School of Reliability and Systems Engineering Beihang University,
More informationA New Uncertain Programming Model for Grain Supply Chain Design
INFORMATION Volume 5, Number, pp.-8 ISSN 343-4500 c 0 International Information Institute A New Uncertain Programming Model for Grain Supply Chain Design Sibo Ding School of Management, Henan University
More informationSome limit theorems on uncertain random sequences
Journal of Intelligent & Fuzzy Systems 34 (218) 57 515 DOI:1.3233/JIFS-17599 IOS Press 57 Some it theorems on uncertain random sequences Xiaosheng Wang a,, Dan Chen a, Hamed Ahmadzade b and Rong Gao c
More informationOn the convergence of uncertain random sequences
Fuzzy Optim Decis Making (217) 16:25 22 DOI 1.17/s17-16-9242-z On the convergence of uncertain random sequences H. Ahmadzade 1 Y. Sheng 2 M. Esfahani 3 Published online: 4 June 216 Springer Science+Business
More informationFormulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable
1 Formulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable Xiumei Chen 1,, Yufu Ning 1,, Xiao Wang 1, 1 School of Information Engineering, Shandong Youth University of Political
More informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationAn Analytic Method for Solving Uncertain Differential Equations
Journal of Uncertain Systems Vol.6, No.4, pp.244-249, 212 Online at: www.jus.org.uk An Analytic Method for Solving Uncertain Differential Equations Yuhan Liu Department of Industrial Engineering, Tsinghua
More informationUncertain Models on Railway Transportation Planning Problem
Uncertain Models on Railway Transportation Planning Problem Yuan Gao, Lixing Yang, Shukai Li State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University Beijing 100044, China Abstract
More informationAccepted Manuscript. Uncertain Random Assignment Problem. Sibo Ding, Xiao-Jun Zeng
Accepted Manuscript Uncertain Random Assignment Problem Sibo Ding, Xiao-Jun Zeng PII: S0307-904X(17)30717-5 DOI: 10.1016/j.apm.2017.11.026 Reference: APM 12068 To appear in: Applied Mathematical Modelling
More informationUncertain Programming Model for Solid Transportation Problem
INFORMATION Volume 15, Number 12, pp.342-348 ISSN 1343-45 c 212 International Information Institute Uncertain Programming Model for Solid Transportation Problem Qing Cui 1, Yuhong Sheng 2 1. School of
More informationRunge-Kutta Method for Solving Uncertain Differential Equations
Yang and Shen Journal of Uncertainty Analysis and Applications 215) 3:17 DOI 1.1186/s4467-15-38-4 RESEARCH Runge-Kutta Method for Solving Uncertain Differential Equations Xiangfeng Yang * and Yuanyuan
More informationTHE inverse shortest path problem is one of the most
JOURNAL OF NETWORKS, VOL. 9, NO. 9, SETEMBER 204 2353 An Inverse Shortest ath roblem on an Uncertain Graph Jian Zhou, Fan Yang, Ke Wang School of Management, Shanghai University, Shanghai 200444, China
More informationA MODEL FOR THE INVERSE 1-MEDIAN PROBLEM ON TREES UNDER UNCERTAIN COSTS. Kien Trung Nguyen and Nguyen Thi Linh Chi
Opuscula Math. 36, no. 4 (2016), 513 523 http://dx.doi.org/10.7494/opmath.2016.36.4.513 Opuscula Mathematica A MODEL FOR THE INVERSE 1-MEDIAN PROBLEM ON TREES UNDER UNCERTAIN COSTS Kien Trung Nguyen and
More informationA Chance-Constrained Programming Model for Inverse Spanning Tree Problem with Uncertain Edge Weights
A Chance-Constrained Programming Model for Inverse Spanning Tree Problem with Uncertain Edge Weights 1 Xiang Zhang, 2 Qina Wang, 3 Jian Zhou* 1, First Author School of Management, Shanghai University,
More informationOptimizing Project Time-Cost Trade-off Based on Uncertain Measure
INFORMATION Volume xx, Number xx, pp.1-9 ISSN 1343-45 c 21x International Information Institute Optimizing Project Time-Cost Trade-off Based on Uncertain Measure Hua Ke 1, Huimin Liu 1, Guangdong Tian
More informationUncertain Structural Reliability Analysis
Uncertain Structural Reliability Analysis Yi Miao School of Civil Engineering, Tongji University, Shanghai 200092, China 474989741@qq.com Abstract: The reliability of structure is already applied in some
More informationUncertain Quadratic Minimum Spanning Tree Problem
Uncertain Quadratic Minimum Spanning Tree Problem Jian Zhou Xing He Ke Wang School of Management Shanghai University Shanghai 200444 China Email: zhou_jian hexing ke@shu.edu.cn Abstract The quadratic minimum
More informationAn Uncertain Bilevel Newsboy Model with a Budget Constraint
Journal of Uncertain Systems Vol.12, No.2, pp.83-9, 218 Online at: www.jus.org.uk An Uncertain Bilevel Newsboy Model with a Budget Constraint Chunliu Zhu, Faquan Qi, Jinwu Gao School of Information, Renmin
More informationNetwork Flows. 4. Maximum Flows Problems. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 4. Maximum Flows Problems 4.1 Introduction Fall 2010 Instructor: Dr. Masoud Yaghini Introduction Introduction The maximum flow problem In a capacitated network, we wish
More informationSensitivity and Stability Analysis in Uncertain Data Envelopment (DEA)
Sensitivity and Stability Analysis in Uncertain Data Envelopment (DEA) eilin Wen a,b, Zhongfeng Qin c, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b Department of System
More informationStructural Reliability Analysis using Uncertainty Theory
Structural Reliability Analysis using Uncertainty Theory Zhuo Wang Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 00084, China zwang058@sohu.com Abstract:
More informationUncertain Systems are Universal Approximators
Uncertain Systems are Universal Approximators Zixiong Peng 1 and Xiaowei Chen 2 1 School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, China 2 epartment of Risk Management
More informationUncertain Distribution-Minimum Spanning Tree Problem
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 24, No. 4 (2016) 537 560 c World Scientific Publishing Company DOI: 10.1142/S0218488516500264 Uncertain Distribution-Minimum
More informationChance Order of Two Uncertain Random Variables
Journal of Uncertain Systems Vol.12, No.2, pp.105-122, 2018 Online at: www.jus.org.uk Chance Order of Two Uncertain andom Variables. Mehralizade 1, M. Amini 1,, B. Sadeghpour Gildeh 1, H. Ahmadzade 2 1
More informationOn Liu s Inference Rule for Uncertain Systems
On Liu s Inference Rule for Uncertain Systems Xin Gao 1,, Dan A. Ralescu 2 1 School of Mathematics Physics, North China Electric Power University, Beijing 102206, P.R. China 2 Department of Mathematical
More informationInternet Routing Example
Internet Routing Example Acme Routing Company wants to route traffic over the internet from San Fransisco to New York. It owns some wires that go between San Francisco, Houston, Chicago and New York. The
More informationAlgorithms and Theory of Computation. Lecture 11: Network Flow
Algorithms and Theory of Computation Lecture 11: Network Flow Xiaohui Bei MAS 714 September 18, 2018 Nanyang Technological University MAS 714 September 18, 2018 1 / 26 Flow Network A flow network is a
More informationGraphs and Network Flows IE411. Lecture 15. Dr. Ted Ralphs
Graphs and Network Flows IE411 Lecture 15 Dr. Ted Ralphs IE411 Lecture 15 1 Preflow-Push Algorithms First developed by A. V. Goldberg in 1985. Best preflow-push algorithms outperform best augmenting path
More informationUncertain flexible flow shop scheduling problem subject to breakdowns
Journal of Intelligent & Fuzzy Systems 32 (2017) 207 214 DOI:10.3233/JIFS-151400 IOS Press 207 Uncertain flexible flow shop scheduling problem subject to breakdowns Jiayu Shen and Yuanguo Zhu School of
More informationUncertain Entailment and Modus Ponens in the Framework of Uncertain Logic
Journal of Uncertain Systems Vol.3, No.4, pp.243-251, 2009 Online at: www.jus.org.uk Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic Baoding Liu Uncertainty Theory Laboratory
More informationDistance-based test for uncertainty hypothesis testing
Sampath and Ramya Journal of Uncertainty Analysis and Applications 03, :4 RESEARCH Open Access Distance-based test for uncertainty hypothesis testing Sundaram Sampath * and Balu Ramya * Correspondence:
More informationSolution of Fuzzy Maximal Flow Network Problem Based on Generalized Trapezoidal Fuzzy Numbers with Rank and Mode
International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 9, Issue 7 (January 2014), PP. 40-49 Solution of Fuzzy Maximal Flow Network Problem
More informationTitle: Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem
Title: Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem Author: Lin Chen Jin Peng Bo Zhang PII: S1568-4946(16)3596-8 DOI: http://dx.doi.org/doi:1.116/j.asoc.216.11.27 Reference:
More informationUncertain Logic with Multiple Predicates
Uncertain Logic with Multiple Predicates Kai Yao, Zixiong Peng Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 100084, China yaok09@mails.tsinghua.edu.cn,
More informationUNCERTAIN OPTIMAL CONTROL WITH JUMP. Received December 2011; accepted March 2012
ICIC Express Letters Part B: Applications ICIC International c 2012 ISSN 2185-2766 Volume 3, Number 2, April 2012 pp. 19 2 UNCERTAIN OPTIMAL CONTROL WITH JUMP Liubao Deng and Yuanguo Zhu Department of
More informationDiscrete Optimization 2010 Lecture 3 Maximum Flows
Remainder: Shortest Paths Maximum Flows Discrete Optimization 2010 Lecture 3 Maximum Flows Marc Uetz University of Twente m.uetz@utwente.nl Lecture 3: sheet 1 / 29 Marc Uetz Discrete Optimization Outline
More informationMathematics for Decision Making: An Introduction. Lecture 13
Mathematics for Decision Making: An Introduction Lecture 13 Matthias Köppe UC Davis, Mathematics February 17, 2009 13 1 Reminder: Flows in networks General structure: Flows in networks In general, consider
More informationON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 18, No. 1 (2010 1 11 c World Scientific Publishing Company DOI: 10.1142/S0218488510006349 ON LIU S INFERENCE RULE FOR UNCERTAIN
More informationThe Simplest and Smallest Network on Which the Ford-Fulkerson Maximum Flow Procedure May Fail to Terminate
Journal of Information Processing Vol.24 No.2 390 394 (Mar. 206) [DOI: 0.297/ipsjjip.24.390] Regular Paper The Simplest and Smallest Network on Which the Ford-Fulkerson Maximum Flow Procedure May Fail
More informationUncertain Satisfiability and Uncertain Entailment
Uncertain Satisfiability and Uncertain Entailment Zhuo Wang, Xiang Li Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China zwang0518@sohu.com, xiang-li04@mail.tsinghua.edu.cn
More informationFlow Network. The following figure shows an example of a flow network:
Maximum Flow 1 Flow Network The following figure shows an example of a flow network: 16 V 1 12 V 3 20 s 10 4 9 7 t 13 4 V 2 V 4 14 A flow network G = (V,E) is a directed graph. Each edge (u, v) E has a
More informationThe Maximum Flow Problem with Disjunctive Constraints
The Maximum Flow Problem with Disjunctive Constraints Ulrich Pferschy Joachim Schauer Abstract We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative
More informationUncertain Second-order Logic
Uncertain Second-order Logic Zixiong Peng, Samarjit Kar Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Department of Mathematics, National Institute of Technology, Durgapur
More informationPreliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
More informationThe maximum flow problem
The maximum flow problem A. Agnetis 1 Basic properties Given a network G = (N, A) (having N = n nodes and A = m arcs), and two nodes s (source) and t (sink), the maximum flow problem consists in finding
More informationYuefen Chen & Yuanguo Zhu
Indefinite LQ optimal control with equality constraint for discrete-time uncertain systems Yuefen Chen & Yuanguo Zhu Japan Journal of Industrial and Applied Mathematics ISSN 0916-7005 Volume 33 Number
More informationAlgebraic Approaches to Stochastic Optimization
Clemson University TigerPrints All Dissertations Dissertations 5-2012 Algebraic Approaches to Stochastic Optimization Katherine Hastings Clemson University, hasting@clemson.edu Follow this and additional
More informationThe covariance of uncertain variables: definition and calculation formulae
Fuzzy Optim Decis Making 218 17:211 232 https://doi.org/1.17/s17-17-927-3 The covariance of uncertain variables: definition and calculation formulae Mingxuan Zhao 1 Yuhan Liu 2 Dan A. Ralescu 2 Jian Zhou
More informationStrongly Polynomial Algorithm for a Class of Minimum-Cost Flow Problems with Separable Convex Objectives
Strongly Polynomial Algorithm for a Class of Minimum-Cost Flow Problems with Separable Convex Objectives László A. Végh April 12, 2013 Abstract A well-studied nonlinear extension of the minimum-cost flow
More informationUNCORRECTED PROOF. Importance Index of Components in Uncertain Reliability Systems. RESEARCH Open Access 1
Gao and Yao Journal of Uncertainty Analysis and Applications _#####################_ DOI 10.1186/s40467-016-0047-y Journal of Uncertainty Analysis and Applications Q1 Q2 RESEARCH Open Access 1 Importance
More informationLecture 3. 1 Polynomial-time algorithms for the maximum flow problem
ORIE 633 Network Flows August 30, 2007 Lecturer: David P. Williamson Lecture 3 Scribe: Gema Plaza-Martínez 1 Polynomial-time algorithms for the maximum flow problem 1.1 Introduction Let s turn now to considering
More informationThe Minimum Cost Network Flow Problem
EMIS 8374 [MCNFP Review] 1 The Minimum Cost Network Flow Problem Problem Instance: Given a network G = (N, A), with a cost c ij, upper bound u ij, and lower bound l ij associated with each directed arc
More informationWhy is There a Need for Uncertainty Theory?
Journal of Uncertain Systems Vol6, No1, pp3-10, 2012 Online at: wwwjusorguk Why is There a Need for Uncertainty Theory? Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua
More informationA Bound for the Number of Different Basic Solutions Generated by the Simplex Method
ICOTA8, SHANGHAI, CHINA A Bound for the Number of Different Basic Solutions Generated by the Simplex Method Tomonari Kitahara and Shinji Mizuno Tokyo Institute of Technology December 12th, 2010 Contents
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More informationStability and attractivity in optimistic value for dynamical systems with uncertainty
International Journal of General Systems ISSN: 38-179 (Print 1563-514 (Online Journal homepage: http://www.tandfonline.com/loi/ggen2 Stability and attractivity in optimistic value for dynamical systems
More informationEstimating the Variance of the Square of Canonical Process
Estimating the Variance of the Square of Canonical Process Youlei Xu Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 184, China uyl1@gmail.com Abstract Canonical
More informationLecture 8 Network Optimization Algorithms
Advanced Algorithms Floriano Zini Free University of Bozen-Bolzano Faculty of Computer Science Academic Year 2013-2014 Lecture 8 Network Optimization Algorithms 1 21/01/14 Introduction Network models have
More informationTheoretical Foundation of Uncertain Dominance
Theoretical Foundation of Uncertain Dominance Yang Zuo, Xiaoyu Ji 2 Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 84, China 2 School of Business, Renmin
More informationMaximum flow problem CE 377K. February 26, 2015
Maximum flow problem CE 377K February 6, 05 REVIEW HW due in week Review Label setting vs. label correcting Bellman-Ford algorithm Review MAXIMUM FLOW PROBLEM Maximum Flow Problem What is the greatest
More informationHybrid Logic and Uncertain Logic
Journal of Uncertain Systems Vol.3, No.2, pp.83-94, 2009 Online at: www.jus.org.uk Hybrid Logic and Uncertain Logic Xiang Li, Baoding Liu Department of Mathematical Sciences, Tsinghua University, Beijing,
More informationMaximum Flow. Jie Wang. University of Massachusetts Lowell Department of Computer Science. J. Wang (UMass Lowell) Maximum Flow 1 / 27
Maximum Flow Jie Wang University of Massachusetts Lowell Department of Computer Science J. Wang (UMass Lowell) Maximum Flow 1 / 27 Flow Networks A flow network is a weighted digraph G = (V, E), where the
More information10 Max-Flow Min-Cut Flows and Capacitated Graphs 10 MAX-FLOW MIN-CUT
10 Max-Flow Min-Cut 10.1 Flows and Capacitated Graphs Previously, we considered weighted graphs. In this setting, it was natural to thinking about minimizing the weight of a given path. In fact, we considered
More informationEnergy minimization via graph-cuts
Energy minimization via graph-cuts Nikos Komodakis Ecole des Ponts ParisTech, LIGM Traitement de l information et vision artificielle Binary energy minimization We will first consider binary MRFs: Graph
More informationSpectral Measures of Uncertain Risk
Spectral Measures of Uncertain Risk Jin Peng, Shengguo Li Institute of Uncertain Systems, Huanggang Normal University, Hubei 438, China Email: pengjin1@tsinghua.org.cn lisg@hgnu.edu.cn Abstract: A key
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationMaximum flow problem (part I)
Maximum flow problem (part I) Combinatorial Optimization Giovanni Righini Università degli Studi di Milano Definitions A flow network is a digraph D = (N,A) with two particular nodes s and t acting as
More informationCSC Design and Analysis of Algorithms. LP Shader Electronics Example
CSC 80- Design and Analysis of Algorithms Lecture (LP) LP Shader Electronics Example The Shader Electronics Company produces two products:.eclipse, a portable touchscreen digital player; it takes hours
More informationOn Végh s Strongly Polynomial Algorithm for Generalized Flows
On Végh s Strongly Polynomial Algorithm for Generalized Flows by Venus Hiu Ling Lo A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of
More informationLinear Programming. Jie Wang. University of Massachusetts Lowell Department of Computer Science. J. Wang (UMass Lowell) Linear Programming 1 / 47
Linear Programming Jie Wang University of Massachusetts Lowell Department of Computer Science J. Wang (UMass Lowell) Linear Programming 1 / 47 Linear function: f (x 1, x 2,..., x n ) = n a i x i, i=1 where
More informationCMPSCI 611: Advanced Algorithms
CMPSCI 611: Advanced Algorithms Lecture 12: Network Flow Part II Andrew McGregor Last Compiled: December 14, 2017 1/26 Definitions Input: Directed Graph G = (V, E) Capacities C(u, v) > 0 for (u, v) E and
More informationLecture 2: Network Flows 1
Comp 260: Advanced Algorithms Tufts University, Spring 2011 Lecture by: Prof. Cowen Scribe: Saeed Majidi Lecture 2: Network Flows 1 A wide variety of problems, including the matching problems discussed
More informationGRAPH ALGORITHMS Week 7 (13 Nov - 18 Nov 2017)
GRAPH ALGORITHMS Week 7 (13 Nov - 18 Nov 2017) C. Croitoru croitoru@info.uaic.ro FII November 12, 2017 1 / 33 OUTLINE Matchings Analytical Formulation of the Maximum Matching Problem Perfect Matchings
More informationApproximate Binary Search Algorithms for Mean Cuts and Cycles
Approximate Binary Search Algorithms for Mean Cuts and Cycles S. Thomas McCormick Faculty of Commerce and Business Administration University of British Columbia Vancouver, BC V6T 1Z2 Canada June 1992,
More information22 Max-Flow Algorithms
A process cannot be understood by stopping it. Understanding must move with the flow of the process, must join it and flow with it. The First Law of Mentat, in Frank Herbert s Dune (965) There s a difference
More informationElliptic entropy of uncertain random variables
Elliptic entropy of uncertain random variables Lin Chen a, Zhiyong Li a, Isnaini osyida b, a College of Management and Economics, Tianjin University, Tianjin 372, China b Department of Mathematics, Universitas
More informationAn Efficient Algorithm for Computing Robust Minimum Capacity s-t Cuts
An Efficient Algorithm for Computing Robust Minimum Capacity s-t Cuts Doug Altner H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology 765 Ferst Drive NW, Atlanta,
More informationA NEW PROPERTY AND A FASTER ALGORITHM FOR BASEBALL ELIMINATION
A NEW PROPERTY AND A FASTER ALGORITHM FOR BASEBALL ELIMINATION KEVIN D. WAYNE Abstract. In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team
More informationNetwork Flows. CTU FEE Department of control engineering. March 28, 2017
Network Flows Zdeněk Hanzálek, Přemysl Šůcha hanzalek@fel.cvut.cz CTU FEE Department of control engineering March 28, 2017 Z. Hanzálek (CTU FEE) Network Flows March 28, 2017 1 / 44 Table of contents 1
More informationMathematical Programming Involving (α, ρ)-right upper-dini-derivative Functions
Filomat 27:5 (2013), 899 908 DOI 10.2298/FIL1305899Y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Mathematical Programming Involving
More information1 Review for Lecture 2 MaxFlow
Comp 260: Advanced Algorithms Tufts University, Spring 2009 Prof. Lenore Cowen Scribe: Wanyu Wang Lecture 13: Back to MaxFlow/Edmonds-Karp 1 Review for Lecture 2 MaxFlow A flow network showing flow and
More informationAgenda. Soviet Rail Network, We ve done Greedy Method Divide and Conquer Dynamic Programming
Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Now Flow Networks, Max-flow Min-cut and Applications c Hung Q. Ngo (SUNY at Buffalo) CSE 531 Algorithm Analysis and Design 1 / 52
More informationdirected weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time
Network Flow 1 The Maximum-Flow Problem directed weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time 2 Maximum Flows and Minimum Cuts flows and cuts max flow equals
More informationThe max flow problem. Ford-Fulkerson method. A cut. Lemma Corollary Max Flow Min Cut Theorem. Max Flow Min Cut Theorem
The max flow problem Ford-Fulkerson method 7 11 Ford-Fulkerson(G) f = 0 while( simple path p from s to t in G f ) 10-2 2 1 f := f + f p output f 4 9 1 2 A cut Lemma 26. + Corollary 26.6 Let f be a flow
More informationORIE 6334 Spectral Graph Theory October 25, Lecture 18
ORIE 6334 Spectral Graph Theory October 25, 2016 Lecturer: David P Williamson Lecture 18 Scribe: Venus Lo 1 Max Flow in Undirected Graphs 11 Max flow We start by reviewing the maximum flow problem We are
More information