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Minimum Cost Circulation Problem
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 (i.e. there are no special nodes). In variants of the problem, there are multiple commodities flowing through the network, and a cost on the flow. Definition Given flow network G(V,E) with: :l(v,w), lower bound on flow from node v to node w, :u(v,w), upper bound on flow from node v to node w, :c(v,w), cost of a unit of flow on (v,w) and the constraints: :l(v,w) \leq f(v,w) \leq u(v,w), :\sum_ f(u,w) = 0 (flow cannot appear or disappear in nodes). Finding a flow assignment satisfying the constraints gives a solution to the given circulation problem. In the minimum cost variant of the problem, minimize : \sum_ c(v,w) \cdot f(v,w). Multi-commodity circulation In a multi-commodity circulation problem, you also need to keep track of the flow of the individual com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Network
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 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. Definition A network is a graph , where is a set of vertices and is a set of 's edges – a subset of – together with a non-negative function , called the capacity function. Without loss of generality, we may assume that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmonds–Karp Algorithm
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in O(, V, , E, ^2) time. The algorithm was first published by Yefim Dinitz (whose name is also transliterated "E. A. Dinic", notably as author of his early papers) in 1970 and independently published by Jack Edmonds and Richard Karp in 1972. Dinic's algorithm includes additional techniques that reduce the running time to O(, V, ^2, E, ). Algorithm The algorithm is identical to the Ford–Fulkerson algorithm, except that the search order when finding the augmenting path is defined. The path found must be a shortest path that has available capacity. This can be found by a breadth-first search, where we apply a weight of 1 to each edge. The running time of O(, V, , E, ^2) is found by showing that each augmenting path can be found in O(, E, ) time, that every time at least one of the E edges becomes saturated (an edge which has the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
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 by linear function#As a polynomial function, 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 mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-commodity Flow Problem
The multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes. Definition Given a flow network \,G(V,E), where edge (u,v) \in E has capacity \,c(u,v). There are \,k commodities K_1,K_2,\dots,K_k, defined by \,K_i=(s_i,t_i,d_i), where \,s_i and \,t_i is the source and sink of commodity \,i, and \,d_i is its demand. The variable \,f_i(u,v) defines the fraction of flow \,i along edge \,(u,v), where \,f_i(u,v) \in ,1/math> in case the flow can be split among multiple paths, and \,f_i(u,v) \in \ otherwise (i.e. "single path routing"). Find an assignment of all flow variables which satisfies the following four constraints: (1) Link capacity: The sum of all flows routed over a link does not exceed its capacity. :\forall (u,v)\in E:\,\sum_^ f_i(u,v)\cdot d_i \leq c(u,v) (2) Flow conservation on transit nodes: The amount of a flow entering an intermediate node u is the same that exits the node. :\forall i \in K: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Cost Multi-commodity Flow Problem
The multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes. Definition Given a flow network \,G(V,E), where edge (u,v) \in E has capacity \,c(u,v). There are \,k commodities K_1,K_2,\dots,K_k, defined by \,K_i=(s_i,t_i,d_i), where \,s_i and \,t_i is the source and sink of commodity \,i, and \,d_i is its demand. The variable \,f_i(u,v) defines the fraction of flow \,i along edge \,(u,v), where \,f_i(u,v) \in ,1/math> in case the flow can be split among multiple paths, and \,f_i(u,v) \in \ otherwise (i.e. "single path routing"). Find an assignment of all flow variables which satisfies the following four constraints: (1) Link capacity: The sum of all flows routed over a link does not exceed its capacity. :\forall (u,v)\in E:\,\sum_^ f_i(u,v)\cdot d_i \leq c(u,v) (2) Flow conservation on transit nodes: The amount of a flow entering an intermediate node u is the same that exits the node. :\forall i \in K: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Cost Flow Problem
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. Definition A flow network is a directed graph G=(V,E) with a source vertex s \in V and a sink vertex t \in V, where each edge (u,v) \in E has capacity c(u,v) > 0, flow f(u,v) and cost a(u,v), with most minimum-cost flow algorithms supporting edges with negative costs. The cost of sending this flow along an edge (u,v) is f(u,v)\cdot a(u,v). The problem requires an amount of flow d to be sent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Flow Problem
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 case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. History The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962). In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):Consider a rail network conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Cost Maximum Flow Problem
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. Definition A flow network is a directed graph G=(V,E) with a source vertex s \in V and a sink vertex t \in V, where each edge (u,v) \in E has capacity c(u,v) > 0, flow f(u,v) and cost a(u,v), with most minimum-cost flow algorithms supporting edges with negative costs. The cost of sending this flow along an edge (u,v) is f(u,v)\cdot a(u,v). The problem requires an amount of flow d to be sent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path Problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \leq i ... [...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] |
