Max-flow min-cut theorem

This paper aims to optimize the content-aware prioritization of scalable video multicast, which is coupled with multipath streaming and network coding based routing. It constructs multiple layer distribution meshes for the scalable video stream to minimize the total video distortion at all the receivers, determines the base layer meshes with minimum costs to maintain application-layer QoS and the layer synchronization of SVC streaming, and improves the network throughput by encouraging path-overlapping transmissions and thus allowing bandwidth sharing among different receivers for the same video layer by utilizing network coding. The targeted problem is formulated into a minimization programming in which the quality variation between layers, the transmission cost of the base layer, as well as the efficient resource utilization are jointly considered. By decomposition and dual approach, the global convex problem is solved by a two-level decentralized iterative algorithm. The implementation of the distributed algorithm is discussed with regard to the communication overhead, and the convergence performance is validated by numerical experiments. Packet-level simulations demonstrate that the proposed algorithm could approximately achieve the maximum flow rates determined by Max-Flow Min-Cut Theorem and benefit the overall received video quality.

Polymatroidal flows with lower bounds

This paper proposes a maximum-flow-based complex network approach for the analysis of the vulnerability of power systems. A new centrality index is proposed, taking into consideration the maximum flow from the source (generator) nodes to the sink (load) nodes, for assessing the network. The Max-Flow Min-Cut Theorem, also known as Ford-Fulkerson Theorem, is used for evaluating the capacity of links. The proposed methodology is then used to identify vulnerable lines of the IEEE 118 bus system and its effectiveness is demonstrated through simulation studies.

The literature of combinatorial optimization, and of related fields such as graph theory. is replete with numerous examples of duality theorems and their applications: the Max-Flow Min-Cut Theorem, Konig-Egervary Theorem, Edmond’s Odd Set Covering Theorem, Hoffman’s Circulation Theorem, Minty’s Painting Theorem, Menger’s Theorem, Dilworth’s Theorem, etc. Our objective in this Chapter is to review a few of these duality results and to give some examples which may be convincing of the usefulness of duality concepts in problem formulation and solution.

The aim is to give rigorous formulations of max-flow problems and min-cut problems an a bounded domain Ω in R n with Lipschitz boundary and to prove max-flow min-cut theorems. For example, one has to give a rigorous notion of a flow σ, the outer unit normal v, the inner product σ.v and the cut capacity of a cut. To do so, one basically follows Strang's idea. The space of essentially bounded vector fields with divergence in L n (Ω) and the space of functions of bounded variations will play important roles in this study. In fact, a static flow is represented by an element in the former space and a static cut is represented by a characteristic function which belongs to the latter space. One of the mathematical tools is a generalized Greens' formula, for functions of bounded variation and essentially bounded vector fields with divergence in L n (Ω). Furthermore one treats the general case when the network is anisotropic

We introduce polylinking networks, which is a flow model based on polylinking systems that generalizes the classical flow model of Ford and Fulkerson on acyclic networks and has applications in the context of wireless networks. More precisely, a flow model recently introduced by Avestimehr et al. (Proceedings of the Allerton conference on communication, control, and computing 2007) used in the context of wireless information networks is a special case of the presented model. We define a notion of source-destination cut and derive a max-flow min-cut theorem. Additionally, we present various properties of polylinking networks that can be seen as generalizations of properties for classical flows. Using submodular function minimization and submodular flow algorithms, one can efficiently determine a maximum flow, a minimum source-destination cut, as well as a minimum cost flow. These algorithms lead to new efficient algorithms for the information flow model.

In this paper, the algorithm of network maximum flow based on network flow matrix is introduced first, then some new properties of network flow matrix are studied according to max-flow min-cut theorem, and on this basis the improved algorithm of network maximum flow based on network flow matrix is given. In this way, the matrix can be reduced order directly when it satisfies certain particular conditions and be translated into several matrices when its order is large, thus simplifying the original algorithm.KeywordsNetwork Flow MatrixNetwork Maximum FlowMax-flow Min-cut Theorem

The problem of network coding for multicasting a single source to multiple sinks has first been studied by Ahlswede, Cai, Li, and Yeung in 2000, in which they have established the celebrated max-flow min-cut theorem on nonphysical information flow over a network of independent channels. On the other hand, in 1980, Han has studied the case with multiple correlated sources and a single sink from the viewpoint of polymatroidal functions in which a necessary and sufficient condition has been demonstrated for reliable transmission over the network. This paper presents an attempt to unify both cases, which leads to establish a necessary and sufficient condition for reliable transmission over a network for multicasting multiple correlated sources to multiple sinks. Here, the problem of separation of source coding and network coding is also discussed.

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions.

In this letter, we present a new wireless channel induced coding (WiCiC) scheme over the binary field in order to eliminate the performance loss which occurs due to not taking the channel conditions into account while determining network code coefficients. We utilize the composite channel-state matrix composed of source-to-relay and relay-to-destination channel-state matrices, as a result of the transition from the tripartite graph model to the bipartite graph model of the network. Through the use of this composite channel-state matrix, the encoding problem can be handled as a maximum flow problem owing to the max-flow min-cut theorem. The proposed WiCiC scheme has a lower decoding complexity and decoding delay compared with the benchmark studies while achieving the optimum decoding failure performance and providing user fairness. Extensive simulation results are presented to make reasonable inferences about the decoding performance of the WiCiC scheme.

A multicommodity flow problem is considered where for each pair of vertices (u, v) it is required to send f half-units of commodity (u, v) from u to v and f half-units of commodity (v, u) from v to u without violating capacity constraints. The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor. The condition on cuts is required in the worst case, and is trivially within a Theta (log n) factor of optimal for any flow problem. The result can be used to construct the first polylog-times optimal approximation algorithms for a wide variety of problems, including minimum quotient separators, 1/3-2/3 separators, bifurcators, crossing number, and VLSI layout area. It can also be used to route packets efficiently in arbitrary distributed networks. >

The absolute permeability of porous media is an important parameter for various technological applications ranging from ground water hydrology to hydrocarbon recovery to microfluidics. We investigate the max-flow min-cut theorem which states that the maximum flow through any network is exactly the sum of the edge weights that define the minimum cut. We hypothesize that the min-cut can be related to the absolute permeability of a network. We find correspondence between the absolute permeability measured from pore network modeling (${K}_{p\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}m}$) to that measured from the min-cut (${K}_{e\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}m}$) and provide a means to identify structural regions that result in significant energy dissipation.

Aharoni et al. [Ron Aharoni et al., 2010] proved the max-flow min-cut theorem for countable networks, namely that in every countable network with finite edge capacities, there exists a flow and a cut such that the flow saturates all outgoing edges of the cut and is zero on all incoming edges. In this paper, we formalize their proof in Isabelle/HOL and thereby identify and fix several problems with their proof. We also provide a simpler proof for networks where the total outgoing capacity of all vertices other than the source is finite. This proof is based on the max-flow min-cut theorem for finite networks.

Optimum Synthesis of a Network with Non-Simultaneous Multicommodity Flow Requirements

Network coding technology is a new communication paradigm that is superior to traditional routing in many aspects, especially in terms of the ability of increasing multicast throughput (Ahlswede et al., 2000; Li et al., 2003). Traditional routing adopts store-andforward data forwarding scheme at every intermediate node that simply replicates and forwards the incoming data to downstream nodes. However, the maximum throughput of a multicast scenario could not be often achieved under such a scheme (Ahlswede et al., 2000; Li et al., 2003). With code-and-forward data forwarding scheme at network layer, network coding allows arbitrary intermediate node to combine (or code) the data received from different incoming links and output the coded information if necessary, being able to obtain a multicast throughput that is maximized according to the MAX-FLOW MIN-CUT theorem (Li et al., 2003). Fig. 1 shows why network coding performs better than traditional routing in terms of the maximum multicast throughput they achieve. Fig.1(a) shows a network with source s and sinks y, z. Each direct link has a capacity of 1 bit per time unit. Source s expects to send two bits, a and b, to y and z. According to the MAX-FLOW MIN-CUT theorem, the min cut Cmin between s and {y, z} is 2 bits per time unit, which means the maximum multicast throughput from s to y and z should be 2 bits per time unit. However, if traditional routing is adopted, the multicast throughput is 1.5 bits information per time unit since link wax could only forward 1 bit, a or b, to x, and thus y and z can not simultaneously receive two bits, a and b, as indicated in Fig.1(b). In Fig.1(c), if the intermediate node w is allowed to combine the 2 bits, a and b, it receives from t and u respectively to 1 bit a⊕b (here, symbol ⊕ is ExclusiveOR operation) and output a⊕b to x, y and z are both able to obtain {a, a⊕b} and {b, a⊕b}, which means two bits information is available at both y and z. Meanwhile, y and z can use {a, a⊕b} and {b, a⊕b} to decode b and a by calculate a⊕(a⊕b) and b⊕(a⊕b) respectively. To the best of our knowledge, most of the network-coding-related research works suppose that coding operation should be implemented at all coding-possible intermediate nodes. However, to achieve a desired throughput, coding operation may only be necessarily performed at a subset of those nodes (Kim et al., 2006; 2007a; 2007b). In Fig.2, there are two network coding schemes that could both achieve the maximum multicast throughput. Network coding scheme A adopts all coding-possible nodes, m and n, as shown in Fig.2(a).