In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals.
Specific types of network flow problems include:
The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals and into the sink terminals
The minimum-cost flow problem, in which the edges have costs as well as capacities and the goal is to achieve a given amount of flow (or a maximum flow) that has the minimum possible cost
The multi-commodity flow problem, in which one must construct multiple flows for different commodities whose total flow amounts together respect the capacities
Nowhere-zero flow, a type of flow studied in combinatorics in which the flow amounts are restricted to a finite set of nonzero values
The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum cut, a partition of the vertices of the flow network that minimizes the total capacity of edges crossing from one side of the partition to the other. Approximate max-flow min-cut theorems provide an extension of this result to multi-commodity flow problems. The Gomory–Hu tree of an undirected flow network provides a concise representation of all minimum cuts between different pairs of terminal vertices.
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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 the minimum cut, i.e. the smallest total weight of the edges which if removed would disconnect the source from the sink.
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.
In graph theory, a flow 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 the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes.
In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if we have:
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements that need to be removed to separate the remaining nodes into isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.
In graph theory, a minimum cut or min-cut of a graph is a cut that is minimal in some sense.
In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed.
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs.
The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. The minimum cost flow problem is one of the most fundamental among all flow and circulation problems because most other such problems can be cast as a minimum cost flow problem and also that it can be solved efficiently using the network simplex algorithm.
The circulation problem and its variants are a generalisation of network flow problems, with the added constraint of a lower bound on edge flows, and with flow conservation also being required for the source and sink. In variants of the problem, there are multiple commodities flowing through the network, and a cost on the flow.
In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices with no outgoing edges. That is, the graph should have no edges that start within the closure and end outside the closure. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in polynomial time using a reduction to the maximum flow problem. It may be used to model various application problems of choosing an optimal subset of tasks to perform, with dependencies between pairs of tasks, one example being in open pit mining.
In combinatorial optimization, the Gomory–Hu tree of an undirected graph with capacities is a weighted tree that represents the minimum s-t cuts for all s-t pairs in the graph. The Gomory–Hu tree can be constructed in | V | − 1 maximum flow computations.
In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.
The HCS clustering algorithm is an algorithm based on graph connectivity for cluster analysis. It works by representing the similarity data in a similarity graph, and then finding all the highly connected subgraphs. It does not make any prior assumptions on the number of the clusters. This algorithm was published by Erez Hartuv and Ron Shamir in 1998.
Approximate max-flow min-cut theorems are mathematical propositions in network flow theory. They deal with the relationship between maximum flow rate ("max-flow") and minimum cut ("min-cut") in a multi-commodity flow problem. The theorems have enabled the development of approximation algorithms for use in graph partition and related problems.
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