# Algorithm Movie Sub Indo 49 !!LINK!!

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.

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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, many 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%.[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 such problems, the TSP can be embedded inside an optimal control problem. In many applications, additional constraints such as limited resources or time windows may be imposed.

The TSP was mathematically formulated in the 19th century by the Irish mathematician William Rowan Hamilton and by the British mathematician Thomas Kirkman. Hamilton's icosian game was a recreational puzzle based on finding a Hamiltonian cycle.[3] 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:

In the 1950s and 1960s, the problem became increasingly popular in scientific circles in Europe and the United States after the RAND Corporation in Santa Monica offered prizes for steps in solving the problem.[5] 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 number 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.[5] As well as cutting plane methods, Dantzig, Fulkerson and Johnson used branch and bound algorithms perhaps for the first time.[5]

In 1976, Christofides and Serdyukov independently of each other made a big advance in this direction:[9] the Christofides-Serdyukov algorithm yields a solution that, in the worst case, is at most 1.5 times longer than the optimal solution. As the algorithm was simple and quick, many hoped it would give way to a near optimal solution method. However, this hope for improvement did not immediately materialize, and Christofides-Serdyukov remained the method with the best worst-case scenario until 2011, when a (very) slightly improved approximation algorithm was developed for the subset of "graphical" TSPs.[10] In 2020 this tiny improvement was extended to the full (metric) TSP.[11][12]

Improving these time bounds seems to be difficult. For example, it has not been determined whether a classical exact algorithm for TSP that runs in time O ( 1.9999 n ) \displaystyle O(1.9999^n) exists.[24] The currently best quantum exact algorithm for TSP due to Ambainis et al. runs in time O ( 1.728 n ) \displaystyle O(1.728^n) . [25]

The algorithm of Christofides and Serdyukov follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching. This gives a TSP tour which is at most 1.5 times the optimal. It was one of the first approximation algorithms, and was in part responsible for drawing attention to approximation algorithms as a practical approach to intractable problems. As a matter of fact, the term "algorithm" was not commonly extended to approximation algorithms until later; the Christofides algorithm was initially referred to as the Christofides heuristic.[9]

This algorithm looks at things differently by using a result from graph theory which helps improve on the lower bound of the TSP which originated from doubling the cost of the minimum spanning tree. Given an Eulerian graph we can find an Eulerian tour in O ( n ) \displaystyle O(n) time.[5] So if we had an Eulerian graph with cities from a TSP as vertices then we can easily see that we could use such a method for finding an Eulerian tour to find a TSP solution. By triangular inequality we know that the TSP tour can be no longer than the Eulerian tour and as such we have a lower bound for the TSP. Such a method is described below.

To improve the lower bound, a better way of creating an Eulerian graph is needed. By triangular inequality, the best Eulerian graph must have the same cost as the best travelling salesman tour, hence finding optimal Eulerian graphs is at least as hard as TSP. One way of doing this is by minimum weight matching using algorithms of O ( n 3 ) \displaystyle O(n^3) .[5]

Making a graph into an Eulerian graph starts with the minimum spanning tree. Then all the vertices of odd order must be made even. So a matching for the odd degree vertices must be added which increases the order of every odd degree vertex by one.[5] This leaves us with a graph where every vertex is of even order which is thus Eulerian. Adapting the above method gives the algorithm of Christofides and Serdyukov.

TSP is a touchstone for many general heuristics devised for combinatorial optimization such as genetic algorithms, simulated annealing, tabu search, ant colony optimization, river formation dynamics (see swarm intelligence) and the cross entropy method.

Like the general TSP, the exact Euclidean TSP is NP-hard, but the issue with sums of radicals is an obstacle to proving that its decision version is in NP, and therefore NP-complete. A discretized version of the problem with distances rounded to integers is NP-complete.[38] With rational coordinates and the actual Euclidean metric, Euclidean TSP is known to be in the Counting Hierarchy,[39] a subclass of PSPACE. With arbitrary real coordinates, Euclidean TSP cannot be in such classes, since there are uncountably many possible inputs. Despite these complications, Euclidean TSP is much easier than the general metric case for approximation.[40] For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O (n log n) time for n points (considerably less than the number of edges). This enables the simple 2-approximation algorithm for TSP with triangle inequality above to operate more quickly.

In general, for any c > 0, where d is the number of dimensions in the Euclidean space, there is a polynomial-time algorithm that finds a tour of length at most (1 + 1/c) times the optimal for geometric instances of TSP in

In the general case, finding a shortest travelling salesman tour is NPO-complete.[51] If the distance measure is a metric (and thus symmetric), the problem becomes APX-complete[52] and the algorithm of Christofides and Serdyukov approximates it within 1.5.[53][54][9]

If the distances are restricted to 1 and 2 (but still are a metric) the approximation ratio becomes 8/7.[55] In the asymmetric case with triangle inequality, up until recently only logarithmic performance guarantees were known.[56] In 2018, a constant factor approximation was developed by Svensson, Tarnawski and Végh.[57] The best current algorithm, by Traub and Vygen, achieves performance ratio of 22 + ε \displaystyle 22+\varepsilon .[58] The best known inapproximability bound is 75/74.[59]

The corresponding maximization problem of finding the longest travelling salesman tour is approximable within 63/38.[60] If the distance function is symmetric, the longest tour can be approximated within 4/3 by a deterministic algorithm[61] and within 1 25 ( 33 + ε ) \displaystyle \tfrac 125(33+\varepsilon ) by a randomized algorithm.[62]

For benchmarking of TSP algorithms, TSPLIB[76] is a library of sample instances of the TSP and related problems is maintained, see the TSPLIB external reference. Many of them are lists of actual cities and layouts of actual printed circuits.

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