Jelena Piplica · Repository of Faculty of Science · 2023
Dijkstra algoritam i Bellman-Ford algoritam su algoritmi koji se koriste za pronalaženje najkraćeg puta u težinskom grafu. Dijkstra algoritam djeluje na pozitivnim težinama bridova i koristi se za pronalaženje najkraćeg puta od jednog početnog vrha do svih ostalih vrhova u grafu. Bellman-Ford algoritam, s druge strane, može se koristiti čak i kada postoje negativne težine bridova, ali ima složenost vremena veću od Dijkstra algoritma. Ford-Fulkerson algoritam koristi se za pronalaženje maksimalnog mrežnog protoka, a temelji se na uvećavajućim putevima. Ovaj algoritam iterativno pronalazi uvećavajuće puteve i uvećava protok dok ne postigne maksimalni protok. Ovi algoritmi su važan alat za rješavanje različitih problema u području grafova i optimizacije.
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Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin · Encyclopedia of Optimization · 2001
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Gaja Velkavrh · 2012
Maximum flow problem is one of the fundamental problems in network flow theory and has been extensively investigated. In this work we describe the development of efficient algorithms for the maximum flow problem and the current situation in this area. For each algorithm, we describe the main idea, which is used to achieve a better time complexity and the relationship with previous algorithms. We did not describe algorithms in more detail, because some, especially more recent, are very complicated. Therefore, we mentioned literature where algorithms are described in more detail.
The problem was first defined by Dantzig (1951) and solved by Ford and Fulkerson (1956), using the augmenting paths method. Since then has been an increasing number of algorithms.
Edmonds and Karp (1968) showed that the algorithm of Ford - Fulkerson has time complexity O(nm^2), if the flow is increased firstly on shortest paths. Independently Dinitz (1970) demonstrated the concept of networks with the shortest paths - multi-layered networks - and reached the time complexity O(n^2 m). This limit was improved by Karzanov (1974), who introduced the concept of preflows in multi-layer networks.
This time complexity was later improved by Goldberg and Tarjan (1986). Their algorithm sends preflow from the nodes that are closest to the sink. Information about the distance is better than a multi-layered network, because it is simpler to understand and easier to use in parallel algorithms. By using excess scaling Orlin and Ahuja (1988) improved Goldberg - Tarjan's algorithm.
With progressive combination of blocking backflow and preflow Waisse (1992) showed the fastest algorithm for the maximum flow problem.
With storage of the list of neighbor nodes heights for each node Pham, Lavallee, Bui and Do (2005) improved Goldberg - Tarjan's algorithm. Currently the fastest approximation algorithm is envisioned by Christiano, Kelner, Mandry and Spielman (2010).
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2013
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SpringerReference · 2011
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Fatima Sapundzhi · Bulgarian Portal for Open Science · 2020
The communication network is made up of nodes and links. It carries traffic where traffic flows from a start node (source) to an end node (sink). In general, a communication network can be represented as: (1) a directed network - the flow is directional from one node to another and the links are considered as directional links; or (2) an undirected network - there is no differentiation between the directions of flow. The aim of the maximal flow problem is to find the maximum flow that can be sent from a specified start node to a specified end node through the edges of the network. The maximum flow problem asks for the largest amount of flow that can be transported from one vertex to another. Network flow modelling is used for traffic engineering of networks. It can help in determining routing decisions
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Ravindra K. Ahuja, J.L. Batra, Sanjeev Gupta · 1981
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Wing-Kai Hon, Yujia Huang, Wangyang Li · Theoretical Computer Science · 2026
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maria Afshari Rad, Hossein Taghizadeh Kakhki · 2012
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Darshan Rajesh Chauhan · 2020
In this thesis, a maximum flow-based network interdiction problem considering uncertainties in arc capacities and interdiction resource consumption is solved. The problem consists of two entities with opposing objectives: the goal of the adversary is to maximize the flow of illicit drugs through the network, while the goal of the interdictor is to minimize the maximum flow by completely interdicting arcs given a specified amount of resources. Lack of complete information about the usage patterns of the transportation network by the adversary results in an uncertain estimate of arc capacity and resources required for interdiction by the interdictor. To account for this uncertainty, a robust optimization framework is utilized, resulting in a Robust Network Interdiction Problem (RNIP). A novel mixed-integer linear program is proposed that solves the RNIP. Three heuristics are proposed to solve RNIP, the first based on Lagrangian Relaxation, the second based on Benders' Decomposition, and the third based on Benders' Decomposition enhanced using the Lagrangian Relaxation presolve. Computational experiments show that the third heuristic performs the best with a final MIP gap of less than 5% and a computational time saving of more than 90% for all the test networks when compared to a state-of-the-art mixed integer program solver. Sensitivity analyses are performed to identify budgets of uncertainty that provide a realistic estimate of the actual maximum flow using a Monte Carlo simulation scheme. Finally, robust decisions are compared to decisions not accounting for any uncertainty to evaluate the value of robustness. It is found that robust decisions can provide fairly accurate estimates of possible actual maximum flows in the network. When the interdiction efforts are significant, robust decisions also lead to a reduction in actual maximum flows, as much as 78% on average for a series of test networks, when compared to decisions not accounting for any uncertainty.
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Amir Beck, Nili Guttmann-Beck · Society for Industrial and Applied Mathematics eBooks · 2025
11.1 ■ Introduction
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Lenka Motyčková · Lecture notes in computer science · 1995
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Isma Bentoumi, Fabio Furini, Ali Ridha Mahjoub et al. · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
The Maximum Flow Blocker Problem (MFBP) is a bi-level optimization problem where the leader is a blocker problem and the follower is a Maximum Flow problem. The MFBP consists in determining a minimum weight subset of arcs, called interdicted arcs, to be removed from the graph such that the remaining maximum flow is no larger than a given threshold. The non-interdicted graph refers to the one induced by the set of arcs which are not interdicted. In telecommunication networks, a major issue consists in detecting and facing anomalies. In this context, the MFBP has a key role in resilience analysis, since it gives the maximum number of anomalies that can occur such that the network continue to be operational. To the best of our knowledge, the MFBP has not been addressed in the literature but a closely related problem, called the Maximum Flow Interdiction Problem (MFIP), has been largely studied. This problem consists in finding a set of interdicted arcs that respect a given budget and such that the maximum flow, remaining in the non-interdicted graph, is minimized. We prove that the two problems, MFBP and MFIP, are equivalent. In other words, one can transform an instance of the MFBP to one of the MFIP and vice versa. Using this relation, we develop a compact integer formulation for the MFBP and obtain some complexity results.
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Bentz, Cédric · HAL (Le Centre pour la Communication Scientifique Directe) · 2006
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Adrian Deaconu · Springer Link (Chiba Institute of Technology)
The problem is to modify the capacities of the arcs from a network so that a given feasible flow becomes a maximum flow and the maximum change of the capacities on arcs is minimum. A very fast O(m⋅log(n)) time complexity algorithm for solving this problem is presented, where m is the number of arcs and n is the number of nodes of the network. The case when both, lower and upper bounds of the flow can be modified so that the given feasible flow becomes a maximum flow is also discussed. The algorithm proposed can be adapted to solve this problem, too. The inverse minimum flow problem considering l∞ norm is also studied.
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Donglei Du, R. Ch · 2005
We consider a two-commodity multiroute maximum flow problem in an undirected network— a generalization of the standard two-commodity maximum flow problem. An efficient combinatorial algorithm, which always guarantees a quarter-integer solution when the capacities are integers, is devised to solve a special case based on a novel extension of the augmentingpath technique. Partial results are reported and the difficulties in applying the augmenting-path technique to the general case are explained.
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Orna Kupferman · EPiC series in computing · 2018
In the traditional maximum-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow into outgoing edges. The problem has been extensively used in order to optimize the performance of networks in numerous application areas. The definition of the problem corresponds to a setting in which the authority has control on all vertices of the network. Today’s computing environment involves parties that should be considered adversarial. We survey recent studies on flow games, which capture settings in which the vertices of the network are owned by different and selfish entities. We start with the case of two players, max (the authority), which aims at maximizing the flow, and min (the hostile environment), which aims at minimizing the flow. We argue that such flow games capture many modern settings, such as partially- controlled pipe or road systems or hybrid software-defined communication networks. We then continue to the special case where all vertices are owned by min. This case captures evacuation scenarios, where the goal is to maximize the flow that is guaranteed to travel in the most unfortunate routing decisions. Finally, we study the general case, of multiple players, each with her own target vertex. In all settings, we study the problems of finding the maximal flows, optimal strategies for the players, as well as stability and equilibrium inefficiency in the case of multi-player games. We discuss additional variants and their applications, and point to several interesting open problems.
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Hayley Iben · 2001
v2V f(u; v) = 0 8u 2 V fs; tg 3. Capacity Constraints f(u; v) c(u; v) 8 u; v 2 V c(u; v) = 0 if (u; v) = 2 E The quantity f(u; v) is the ow from vertex u to vertex v. The value of the ow is given by the total ow out of the source jf j = P v2V f(s; v). This value is also equal to the total ow going into the sink jf j = P v2V f(v; t). The maximum-ow problem can then be stated as nding the ow of maximum value in a given ow network from s to
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Masatoshi Sato, Hisashi Aomori, Mamoru Tanaka · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2009
In advance of network communication society by the internet, the way how to send data fast with a little loss becomes an important transportation problem. A generalized maximum flow algorithm gives the best solution for the transportation problem that which route is appropriated to exchange data. Therefore, the importance of the maximum flow algorithm is growing more and more. In this paper, we propose a Maximum-Flow Neural Network (MF-NN) in which branch nonlinearity has a saturation characteristic and by which the maximum flow problem can be solved with analog high-speed parallel processing. That is, the proposed neural network for the maximum flow problem can be realized by a nonlinear resistive circuit where each connection weight between nodal neurons has a sigmodal or piece-wise linear function. The parallel hardware of the MF-NN will be easily implemented.
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Babeniuk G. · CyberLeninK (CyberLeninka)
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