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Network Flow Problem
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Applications Basic applications of combina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dinic's Algorithm
Dinic's algorithm or Dinitz's algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer scientist Yefim Dinitz. The algorithm runs in O(, V, ^2, E, ) time and is similar to the Edmonds–Karp algorithm, which runs in O(, V, , E, ^2) time, in that it uses shortest augmenting paths. The introduction of the concepts of the ''level graph'' and ''blocking flow'' enable Dinic's algorithm to achieve its performance. History Dinitz invented the algorithm in January 1969, as a master's student in Georgy Adelson-Velsky's group. A few decades later, he would recall: In 1970, Dinitz published a description of the algorithm in Proceedings of the USSR Academy of Sciences, ''Doklady Akademii Nauk SSSR''. In 1974, Shimon Even and (his then Ph.D. student) Alon Itai at the Technion – Israel Institute of Technology, Technion in Haifa were very curious and intrigued by Dinitz's algorithm as well as Alexande ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Algorithms
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. Automated planning Combinatorial algorithms General combinatorial algorithms * Brent's algorithm: finds a cycle in function value iterations using only two iterators * Floyd's cycle-finding algorithm: finds a cycle in function value iterations * Gale–Shapley algorithm: solves the stable matching problem * Pseudorandom number generators (uniformly dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Flow Problem
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Program
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 and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists. Linear programs are problems that can be expressed in standard form as: : \beg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Push–relabel Maximum Flow Algorithm
In mathematical optimization, the push–relabel algorithm (alternatively, preflow–push algorithm) is an algorithm for computing maximum flows in a flow network. The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually converts it into a maximum flow by moving flow locally between neighboring nodes using ''push'' operations under the guidance of an admissible network maintained by ''relabel'' operations. In comparison, the Ford–Fulkerson algorithm performs global augmentations that send flow following paths from the source all the way to the sink. The push–relabel algorithm is considered one of the most efficient maximum flow algorithms. The generic algorithm has a strongly polynomial time complexity, which is asymptotically more efficient than the Edmonds–Karp algorithm. Specific variants of the algorithms achieve even lower time complexities. The variant based on the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
