Posts filled under #travelling

#travelwriter #travel #in

#travelwriter #travel #instatravel #travelgram #tourism #instago #passportready #travelblogger #wanderlust #ilovetravel #writetotravel #instatravelling #instavacation #travelblogger #instapassport #postcardsfromtheworld #traveldeeper #travelstroke #travelling #trip #traveltheworld #igtravel #getaway #travelblog #instago #travelpics #tourist #wanderer #wanderlust #travelphoto travelingram mytravelgram visiting travels travelphotography tagsta_travelbeauty amazing arountheworld touristsolotravel instago ig_worldclub worldcaptures tourism worldplaces worldingram traveller traveler hotel luxuryhotel

"Dentro de veinte aos pro

"Dentro de veinte aos probablemente estars ms decepcionado por las cosas que no hiciste que por las que hiciste. As que suelta las amarras, navega lejos del puerto, atrapa los vientos favorables en tus velas, explora, suea, descubre". - Mark Twain . . Disfruta de la vida, de todos los momentos! ------------------------------------------- #spain #barcelona #live #love #photography #photos #photo #sun #summer #photooftheday #travelling #travelblogger #travel #travels #insta #instagood #instagirl #instago #girl #moment #moments #family #mytravelgram #my #film #instaphoto #instalove #instalive #wonderful #happy

Elazig by Night Last Days

Elazig by Night Last Days am Montag fliegen wir endlich in die Heimat Es ist jedesmal ein sehr aufregendes Gefhl auch wenn man manchmal ungern da ist wegen der extremen Hitze varmi aranizda Hemserim Olann 23 Oraya Gitmeden Sanki Trkiyeye Gitmemisim gibi....das ist Fakt #Elazig #Elazigbynight #gakkoslar #Yazikonak #Elazigcity #Turkey #Vacay #Familia #travel #traveling #TagsForLikes #TFLers #vacation #visiting #instatravel #instago #instagood #trip #holiday #photooftheday #fun #travelling #tourism #tourist #instapassport #instatraveling #mytravelgram #travelgram #travelingram

MoroccanPhotographersInst

MoroccanPhotographersInstagramcommunity Presents Photo Of The Day To participate in daily featured Tag your Pictures with #MoroccanPhotos Photo_Credit : @martapn71 Chefchaouen, el paraso azul de Marruecos. Uno de los lugares que ms ha calado en m y al que seguro volver. I love Chaouen (December 2016) #instatravel #instatravelling #instaviajes #travelling #traveladdiction #traveladdict #igtravel #travelgram #ilovetravel #igerviajero #quetalviajar #phototravel #photos #travelphotos #fotosdeviajes #trip #viaje #travelphotography #travellingaroundtheworld #igtravel #picoftheday #photooftheday #chaouen #photooftheworld #marocco #igersofchaouen #chefchaouen #morocco #marocco #marrakech #photos #thebluepearl

The ancient city of Winch

The ancient city of Winchester has been at the heart of feasting for centuries and now The Running Horse keeps up this tradition by delivering fabulous food to feast your eyes on and titillate your tastebuds. This stylishly restored inn is the perfect venue for a leisurely drink or supper with family and friends and they also have 15 modern and comfortable bedrooms @runninghorseinn @uphambrewery #foodie #walking #dogfriendly #puboftheday #travel #britishpub #travelling #uktravel #britain #caskale #beer #uk #instapub #instatravel #staycation #stayinapub #pubgrub #countryinn #summer #visitengland #visitbritain #holiday #winchester #horses #equestrian #bedroom #luxury #interiordesign

An extract on #travelling

Travelling salesman may also refer to: The travelling salesman problem, a problem in discrete or combinatorial optimization Traveling Salesman (1921 film), a 1921 comedy Travelling Salesman (2012 film), a 2012 intellectual thriller "Traveling Salesmen," the twelfth episode of the third season of the US version of The Office

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 any tour shorter than 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 in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact algorithms are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. The TSP also appears in astronomy, as astronomers observing many sources will want to minimize the time spent moving the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed.

The origins of the travelling salesman problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment. The travelling salesman problem was mathematically formulated in the 1800s by the Irish mathematician W.R. Hamilton and by the British mathematician Thomas Kirkman. Hamiltons Icosian Game was a recreational puzzle based on finding a Hamiltonian cycle. The general form of the TSP appears to have been first studied by mathematicians during the 1930s in Vienna and at Harvard, notably by Karl Menger, who defines the problem, considers the obvious brute-force algorithm, and observes the non-optimality of the nearest neighbour heuristic: We denote by messenger problem (since in practice this question should be solved by each postman, anyway also by many travelers) the task to find, for nitely many points whose pairwise distances are known, the shortest route connecting the points. Of course, this problem is solvable by finitely many trials. Rules which would push the number of trials below the number of permutations of the given points, are not known. The rule that one first should go from the starting point to the closest point, then to the point closest to this, etc., in general does not yield the shortest route. It was first considered mathematically in the 1930s by Merrill Flood who was looking to solve a school bus routing problem. Hassler Whitney at Princeton University introduced the name travelling salesman problem soon after. In the 1950s and 1960s, the problem became increasingly popular in scientific circles in Europe and the USA after the RAND Corporation in Santa Monica offered prizes for steps in solving the problem. Notable contributions were made by George Dantzig, Delbert Ray Fulkerson and Selmer M. Johnson from the RAND Corporation, who expressed the problem as an integer linear program and developed the cutting plane method for its solution. They wrote what is considered the seminal paper on the subject in which with these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. Dantzig, Fulkerson and Johnson, however, speculated that given a near optimal solution we may be able to find optimality or prove optimality by adding a small amount of extra inequalities (cuts). They used this idea to solve their initial 49 city problem using a string model. They found they only needed 26 cuts to come to a solution for their 49 city problem. While this paper did not give an algorithmic approach to TSP problems, the ideas that lay within it were indispensable to later creating exact solution methods for the TSP, though it would take 15 years to find an algorithmic approach in creating these cuts. As well as cutting plane methods, Dantzig, Fulkerson and Johnson used branch and bound algorithms perhaps for the first time. In the following decades, the problem was studied by many researchers from mathematics, computer science, chemistry, physics, and other sciences. In the 1960s however a new approach was created, that instead of seeking optimal solutions, one would produce a solution whose length is provably bounded by a multiple of the optimal length, and in doing so create lower bounds for the problem; these may then be used with branch and bound approaches. One method of doing this was to create a minimum spanning tree of the graph and then double all its edges, which produces the bound that the length of an optimal tour is a most twice the weight of a minimum spanning tree. Christofides made a big advance in this approach of giving an approach for which we know the worst-case scenario. Christofides algorithm given in 1976, at worst is 1.5 times longer than the optimal solution. As the algorithm was so simple and quick, many hoped it would give way to a near optimal solution method. However, until 2011 when it was beaten by less than a billionth of a percent, this remained the method with the best worst-case scenario. Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a mathematical explanation for the apparent computational difficulty of finding optimal tours. Great progress was made in the late 1970s and 1980, when Grtschel, Padberg, Rinaldi and others managed to exactly solve instances with up to 2392 cities, using cutting planes and branch-and-bound. In the 1990s, Applegate, Bixby, Chvtal, and Cook developed the program Concorde that has been used in many recent record solutions. Gerhard Reinelt published the TSPLIB in 1991, a collection of benchmark instances of varying difficulty, which has been used by many research groups for comparing results. In 2006, Cook and others computed an optimal tour through an 85,900-city instance given by a microchip layout problem, currently the largest solved TSPLIB instance. For many other instances with millions of cities, solutions can be found that are guaranteed to be within 2-3% of an optimal tour.

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