Travelling salesman problem - Wikipedia. Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. The travelling salesman problem (TSP) asks the following question: . Thus, it is possible that the worst- caserunning 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 1. 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%. 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. History. A handbook for travelling salesmen from 1. Germany and Switzerland, but contains no mathematical treatment.
Hamilton and by the British mathematician Thomas Kirkman. 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. 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 4. 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).
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They used this idea to solve their initial 4. They found they only needed 2. 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 1. In the 1. 96. 0s however a new approach was created, instead of finding optimal solutions, people tried to instead find the worst solutions and in doing so, created lower bounds for the problem.
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These may then be used with branch and bound approaches. One method of doing this was to create the minimum spanning tree of the graph and then multiply the cost of this by 2. His algorithm given in 1.
As the algorithm was so simple and quick, many hoped it would give way to a near optimal solution method. However, until 2. Karp showed in 1. 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.
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Great progress was made in the late 1. Gr. Gerhard Reinelt published the TSPLIB in 1. In 2. 00. 6, Cook and others computed an optimal tour through an 8. 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. It is a minimization problem starting and finishing at a specified vertex after having visited each other vertex exactly once.
Often, the model is a complete graph (i. If no path exists between two cities, adding an arbitrarily long edge will complete the graph without affecting the optimal tour. Asymmetric and symmetric. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic collisions, one- way streets, and airfares for cities with different departure and arrival fees are examples of how this symmetry could break down. Related problems.
The problem is of considerable practical importance, apart from evident transportation and logistics areas. A classic example is in printed circuit manufacturing: scheduling of a route of the drill machine to drill holes in a PCB. In robotic machining or drilling applications, the . One application is encountered in ordering a solution to the cutting stock problem in order to minimize knife changes. Another is concerned with drilling in semiconductor manufacturing, see e. U. S. Noon and Bean demonstrated that the generalized travelling salesman problem can be transformed into a standard travelling salesman problem with the same number of cities, but a modified distance matrix. The sequential ordering problem deals with the problem of visiting a set of cities where precedence relations between the cities exist.
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A common interview question at Google is how to route data among data processing nodes; routes vary by time to transfer the data, but nodes also differ by their computing power and storage, compunding the problem of where to send data. The travelling purchaser problem deals with a purchaser who is charged with purchasing a set of products. He can purchase these products in several cities, but at different prices and not all cities offer the same products. The objective is to find a route between a subset of the cities, which minimizes total cost (travel cost + purchasing cost) and which enables the purchase of all required products. Integer linear programming formulation. Then TSP can be written as the following integer linear programming problem: min.
The last constraints enforce that there is only a single tour covering all cities, and not two or more disjointed tours that only collectively cover all cities. To prove this, it is shown below (1) that every feasible solution contains only one closed sequence of cities, and (2) that for every single tour covering all cities, there are values for the dummy variables ui. For if we sum all the inequalities corresponding to xij=1. The running time for this approach lies within a polynomial factor of O(n!). Note: Number of permutations: (7- 1)!/2 = 3.
Improving these time bounds seems to be difficult. For example, it has not been determined whether an exact algorithm for TSP that runs in time O(1. Note: The number of permutations is much less than Brute force search. Progressive improvement algorithms which use techniques reminiscent of linear programming. Works well for up to 2. Implementations of branch- and- bound and problem- specific cut generation (branch- and- cut.
This approach holds the current record, solving an instance with 8. Applegate et al. Johnson in 1. The computations were performed on a network of 1.
Rice University and Princeton University (see the Princeton external link). The total computation time was equivalent to 2. MHz Alpha processor. In May 2. 00. 4, the travelling salesman problem of visiting all 2. Sweden was solved: a tour of length approximately 7.
The computation took approximately 1. CPU- years (Cook et al. In April 2. 00. 6 an instance with 8. Concorde TSP Solver, taking over 1. CPU- years, see Applegate et al.
Modern methods can find solutions for extremely large problems (millions of cities) within a reasonable time which are with a high probability just 2. The solution changes as the starting point is changed.
The nearest neighbour (NN) algorithm (a greedy algorithm) lets the salesman choose the nearest unvisited city as his next move. This algorithm quickly yields an effectively short route. For N cities randomly distributed on a plane, the algorithm on average yields a path 2. This is true for both asymmetric and symmetric TSPs (Gutin and Yeo, 2.
A variation of NN algorithm, called Nearest Fragment (NF) operator, which connects a group (fragment) of nearest unvisited cities, can find shorter route with successive iterations. The cycles are then stitched to produce the final tour. Christofides' algorithm for the TSP. This gives a TSP tour which is at most 1.
The Christofides algorithm 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 . Given an Eulerian graph we can find an Eulerian tour in O(n) time. By triangular inequality we know that the TSP tour can be no longer than the Eulerian tour and as such we have a LB for the TSP.
Such a method is described below. But 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 that has been proposed is by the concept of minimum weight matching for the creation of which there exist algorithms of O(n. 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. Now we can adapt the above method to give Christofides' algorithm,Find a minimum spanning tree for the problem.
Create a matching for the problem with the set of cities of odd order. Find an Eulerian tour for this graph. Convert to TSP using shortcuts.
Iterative improvement. This is a special case of the k- opt method. Note that the label Lin.
If we start with an initial solution made with a greedy algorithm, the average number of moves greatly decreases again and is O(n). For random starts however, the average number of moves is O(n log(n)). However whilst in order this is a small increase in size, the initial number of moves for small problems is 1. This is because such 2- opt heuristics exploit `bad' parts of a solution such as crossings.