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Ring Star Problem
The ring star problem (RSP) is a NP-hard problem in combinatorial optimization. In a complete weighted mixed graph, the ring star problem aims to find a minimum cost ring star subgraph formed by a cycle (ring part) and a set of arcs (star part) such that each arc's child node belongs to the cycle and each arc's parent node does not. The costs for the arcs are usually different than the cycle's costs. The cycle must contains at least one node which is called the depot or the root. RSP is a generalization of the traveling salesman problem. When the costs of the arcs are infinite and the ring contains all nodes, the RSP reduces to TSP. Some applications of RSP arise in the context of telecommunications, transports or logistics. Exact formulations RSP was first formulated in 1998. The first MILP for solving RSP was introduced in 2004 alongside valid inequalities that improve the formulation. Several exact formulations have since been introduced in order to solve the Ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Star Problem Solution
(The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a Japanese horror media franchise based on the novel series by Koji Suzuki ** ''Ring'' (film), or ''The Ring'', a 1998 Japanese horror film by Hideo Nakata *** ''The Ring'' (2002 film), an American horror film, remake of the 1998 Japanese film ** ''Ring'' (1995 film), a TV film ** ''Rings'' (2005 film), a short film by Jonathan Liebesman ** ''Rings'' (2017 film), an American horror film * "Ring", a season 3 episode of ''Servant'' (TV series) Gaming * ''Ring'' (video game), 1998 * Rings (''Sonic the Hedgehog''), a collectible in ''Sonic the Hedgehog'' games Literature * ''Ring'' (Baxter novel), a 1994 science fiction novel * ''Ring'' (Alexis novel), a 2021 Canadian novel by André Alexis * ''Ring'' (novel series), a Japanese nov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hardness
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...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] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Graph
In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, is an edge with an orientation and can be denoted as \overrightarrow or (u,v) (note that u is the tail and v is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as uv or ,v/math>. For the purpose of our example we will not be considering loops or multiple edges of mixed graphs. A walk in a mixed graph is a sequence v_0,c_1,v_1,c_2,v_2,\dots,c_k,v_k of vertices and edges/arcs such that for every index i, either c_i=v_v_ is an edge of the graph or c_i=\overrightarrow is an arc of the graph. This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A walk is closed if its first and last vertices are the same, and a closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traveling Salesman Problem
In the theory of computational complexity, the travelling salesman problem (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, the vehicle routing problem and the ring star problem are three generalizations of TSP. The decision version of the TSP (where given a length ''L'', the task is to decide whether the graph has a tour whose length is 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 problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telecommunications
Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of transmission may be divided into communication channels for multiplexing, allowing for a single medium to transmit several concurrent Session (computer science), communication sessions. Long-distance technologies invented during the 20th and 21st centuries generally use electric power, and include the electrical telegraph, telegraph, telephone, television, and radio. Early telecommunication networks used metal wires as the medium for transmitting signals. These networks were used for telegraphy and telephony for many decades. In the first decade of the 20th century, a revolution in wireless communication began with breakthroughs including those made in radio communications by Guglielmo Marconi, who won the 1909 Nobel Prize in Physics. Othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transportation Theory (mathematics)
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.G. Monge. ''Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année'', pages 666–704, 1781. In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection ''Transportation Planning Volume I'' for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space". Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich.L. Kantorovich. ''On the translocation of masses.'' C.R. (Doklady) Acad. Sci. URSS (N.S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistics
Logistics is the part of supply chain management that deals with the efficient forward and reverse flow of goods, services, and related information from the point of origin to the Consumption (economics), point of consumption according to the needs of customers. Logistics management is a component that holds the supply chain together. The resources managed in logistics may include tangible goods such as materials, equipment, and supplies, as well as food and other edible items. In military logistics, it is concerned with maintaining army supply lines with food, armaments, ammunition, and spare parts apart from the transportation of troops themselves. Meanwhile, civil logistics deals with acquiring, moving, and storing raw materials, semi-finished goods, and finished goods. For organisations that provide Waste collection, garbage collection, mail deliveries, Public utility, public utilities, and after-sales services, logistical problems must be addressed. Logistics deals with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0–1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \underset && \mathbf^\mathrm \mathbf\\ & \text && A \mathbf \le \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
