Max-flow min-cut theorem

Summary

In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink. This is a special case of the duality theorem for linear programs and can be used to derive Menger's theorem and the Kőnig–Egerváry theorem. Definitions and statement The theorem equates two quantities: the maximum flow through a network, and the minimum capacity of a cut of the network. To state the theorem, each of these notions must first be defined. Network A network consists of

a finite directed graph N = (V, E), where V denotes the finite set of vertices and E ⊆ V×V is the set of directed edges;
a source s ∈ V and a sink t ∈ V;
a capacity function, which is a mapping c:E\to\R^+ denoted by cuv

Official source

https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem

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In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed

In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special ca

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented

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Combinatorial algorithms for wireless information flow

Aurore Amaudruz, Christina Fragouli

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr, Diggavi and Tse proposed a linear binary deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the binary adjacency matrices describing source-destination cuts. However, since there exists an exponential number of cuts, identifying the capacity through exhaustive search becomes infeasible. In this work, we develop a polynomial time algorithm that discovers the relay encoding strategy to achieve the min-cut value in binary linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between edges to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-bit processing at the intermediate nodes, thereby constructively yielding finite length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.

2009

Information flow decomposiiton for network coding

Christina Fragouli, Emina Soljanin

The famous min-cut, max-flow theorem states that a source node can send a commodity through a network to a sink node at the rate determined by the flow of the min-cut separating the source and the sink. Recently it has been shown that by linear re-encoding at nodes in communications networks, the min-cut rate can be also achieved in multicasting to several sinks. Constructing such coding schemes efficiently is the subject of current research. The main idea in this paper is the identification of structural properties of multicast configurations, by decompositing the information flows into a minimal number of subtrees. This decomposition allows us to show that very different networks are equivalent from the coding point of view, and offers a method to identify such equivalence classes. It also allows us to divide the network coding problem into two almost independent problems: one of graph theory and the other of classical channel coding theory. This approach to network coding enables us to derive tight bounds on the network code alphabet size and calculate the throughput improvement network coding can offer for different configurations. But perhaps the most significant strength of our approach concerns future network coding practice. Namely, we propose algorithms to specify the coding operations at network nodes without the knowledge of the overall network topology. Such decentralized designs facilitate the construction of codes which can easily accommodate future changes in the network, e.g., addition of receivers and loss of links.

Institute of Electrical and Electronics Engineers2006

2006