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3-opt
In optimization, 3-opt is a simple local search algorithm for solving the travelling salesperson problem and related network optimization problems. Compared to the simpler 2-opt algorithm, it is slower but can generate higher-quality solutions. 3-opt analysis involves deleting 3 connections (or edges) in a network (or tour), to create 3 sub-tours. Then the 7 different ways of reconnecting the network are analysed to find the optimum one. This process is then repeated for a different set of 3 connections, until all possible combinations have been tried in a network. A single execution of 3-opt has a time complexity of O(n^3). Iterated 3-opt has a higher time complexity. This is the mechanism by which the 3-opt swap manipulates a given route: def reverse_segment_if_better(tour, i, j, k): """If reversing tour :jwould make the tour shorter, then do it.""" # Given tour ..A-B...C-D...E-F... A, B, C, D, E, F = tour -1 tour tour -1 tour tour -1 tour % len(tour) d0 = d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Travelling Salesperson Problem
The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP. In the theory of computational complexity, the decision version of the TSP (where given a length ''L'', the task is to decide whether the graph has a tour of at most ''L'') belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities. The problem was first formulated in 1930 and is one of the most intensively studied p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-opt
In optimization, 2-opt is a simple local search algorithm for solving the traveling salesman problem. The 2-opt algorithm was first proposed by Croes in 1958, although the basic move had already been suggested by Flood.M. M. Flood, The traveling-salesman problem. Operations Res. 4 (1956), pp., 61-75. The main idea behind it is to take a route that crosses over itself and reorder it so that it does not. A complete 2-opt local search will compare every possible valid combination of the swapping mechanism. This technique can be applied to the traveling salesman problem as well as many related problems. These include the vehicle routing problem (VRP) as well as the capacitated VRP, which require minor modification of the algorithm. Pseudocode Visually, one swap looks like: - A B - - A - B - × > - C D - - C - D - In pseudocode, the mechanism by which the 2-opt swap manipulates a given route is as follows. Here v1 and v2 are the first ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lin–Kernighan Heuristic
In combinatorial optimization, Lin–Kernighan is one of the best heuristic algorithm, heuristics for solving the symmetric travelling salesman problem. It belongs to the class of Local search (optimization), local search algorithms, which take a tour (Hamiltonian cycle) as part of the input and attempt to improve it by searching in the neighbourhood of the given tour for one that is shorter, and upon finding one repeats the process from that new one, until encountering a local minimum. As in the case of the related 2-opt and 3-opt algorithms, the relevant measure of "distance" between two tours is the number of edge (graph theory), edges which are in one but not the other; new tours are built by reassembling pieces of the old tour in a different order, sometimes changing the direction in which a sub-tour is traversed. Lin–Kernighan is adaptive and has no fixed number of edges to replace at a step, but favours small numbers such as 2 or 3. Derivation For a given instance (G,c) of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Optimization
Network, networking and networked may refer to: Science and technology * Network theory, the study of graphs as a representation of relations between discrete objects * Network science, an academic field that studies complex networks Mathematics * Networks, a graph with attributes studied in network theory ** Scale-free network, a network whose degree distribution follows a power law ** Small-world network, a mathematical graph in which most nodes are not neighbors, but have neighbors in common * Flow network, a directed graph where each edge has a capacity and each edge receives a flow Biology * Biological network, any network that applies to biological systems * Ecological network, a representation of interacting species in an ecosystem * Neural network, a network or circuit of neurons Technology and communication * Artificial neural network, a computing system inspired by animal brains * Broadcast network, radio stations, television stations, or other electronic media outlets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Search (optimization)
In computer science, local search is a heuristic method for solving computationally hard optimization problems. Local search can be used on problems that can be formulated as finding a solution maximizing a criterion among a number of candidate solutions. Local search algorithms move from solution to solution in the space of candidate solutions (the ''search space'') by applying local changes, until a solution deemed optimal is found or a time bound is elapsed. Local search algorithms are widely applied to numerous hard computational problems, including problems from computer science (particularly artificial intelligence), mathematics, operations research, engineering, and bioinformatics. Examples of local search algorithms are WalkSAT, the 2-opt algorithm for the Traveling Salesman Problem and the Metropolis–Hastings algorithm. Examples Some problems where local search has been applied are: # The vertex cover problem, in which a solution is a vertex cover of a graph, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ORSA (OR)
The Institute for Operations Research and the Management Sciences (INFORMS) is an international society for practitioners in the fields of operations research (O.R.), management science, and analytics. It was established in 1995 with the merger of the Operations Research Society of America (ORSA) and The Institute of Management Sciences (TIMS). Overview INFORMS promotes greater public awareness, interest, and understanding about the benefits of operations research (O.R.) and analytics and provides a variety of programs and services that support lifelong learning and networking for O.R. and analytics professionals. This includes publishing 16 peer-reviewed journals, hosting numerous conferences and meetings, providing continuing education courses and professional certification, and administering dozens of special-interest communities that help professionals network and collaborate with colleagues from around the world. INFORMS is a member of the International Federation of Operation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristic Algorithms
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using trial and error, a rule of thumb or an educated guess. Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues. When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively crea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
