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The traveling salesman problem ('''TSP''') is a problem in Discrete or Combinatorial Optimization . It is a prominent illustration of a class of problems in Computational Complexity Theory which are hard to solve. PROBLEM STATEMENT Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city exactly once and then returns to the starting city? An equivalent formulation in terms of (where the vertices would represent the cities, the edges would represent the roads, and the weights would be the cost or distance of that road), find a Hamiltonian Cycle with the least weight. It can be shown that the requirement of returning to the starting city does not change the Computational Complexity of the problem. A related problem is the in a Weighted Graph with the minimal length of the longest Edge . 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 "cities" are parts to machine or holes (of different sizes) to drill, and the "cost of travel" includes time for retooling the robot (single machine job sequencing problem). COMPUTATIONAL COMPLEXITY The most direct solution would be to try all the Permutation s (ordered combinations) and see which one is cheapest (using Brute Force ), but given that the number of permutations is ''n''! (the Factorial of the number of cities, ''n''), this solution rapidly becomes impractical. Using the techniques of Dynamic Programming , one can solve the problem exactly in time . Although this is exponential, it is still much better than . See Big O Notation . NP-HARDNESS The problem has been shown to be NP-hard (more precisely, it is complete for the Complexity Class FPNP; see the Function Problem article), and the Decision Problem version ("given the costs and a number ''x'', decide whether there is a roundtrip route cheaper than ''x''") is NP-complete . The Bottleneck Traveling Salesman Problem is also NP-hard. The problem remains NP-hard even for the case when the cities are in the plane with Euclidean Distance s, as well as in a number of other restrictive cases. Removing the condition of visiting each city "only once" does not remove the NP-hardness, since it is easily seen that in the planar case an optimal tour visits cities only once (otherwise, by the Triangle Inequality , a shortcut that skips a repeated visit would decrease the tour length). ALGORITHMS The traditional lines of attack for the NP-hard problems are the following:
For benchmarking of TSP algorithms, TSPLIB 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 Circuit s. Exact algorithms
An exact solution for 15,112 German cities from TSPLIB was found in 2001 using the Cutting-plane Method proposed by George Dantzig , Ray Fulkerson , and Selmer Johnson in 1954 , based on Linear Programming . The computations were performed on a network of 110 processors located at Rice University and Princeton University (see the Princeton external link). The total computation time was equivalent to 22.6 years on a single 500 MHz Alpha Processor . In May 2004, the traveling salesman problem of visiting all 24,978 cities in Sweden was solved: a tour of length approximately 72,500 kilometers was found and it was proven that no shorter tour exists. In March 2005, the traveling salesman problem of visiting all 33,810 points in a circuit board was solved using CONCORDE: a tour of length 66,048,945 units was found and it was proven that no shorter tour exists, the computation took approximately 15.7 CPU years. Heuristics Various Approximation Algorithm s, which "quickly" yield "good" solutions with "high" probability, have been devised. Modern methods can find solutions for extremely large problems (millions of cities) within a reasonable time which are with a high probability just 2-3% away from the optimal solution. Several categories of heuristics are recognized. Constructive heuristics
Iterative improvement
Randomized improvement
TSP is a touchstone for many general heuristics devised for combinatorial optimization: Genetic Algorithm s, Simulated Annealing , Tabu Search , Neural Net s, Ant System .. Special cases Restricted locations
Triangle inequality and the Christofides algorithm A very natural restriction of the TSP is the Triangle Inequality . That is, for any 3 cities A, B and C, the distance between A and C must be at most the distance from A to B plus the distance from B to C. Most natural instances of TSP satisfy this constraint. In this case, there is a Constant-factor Approximation Algorithm (due to Christofides, 1975) which always finds a tour of length at most 1.5 times the shortest tour. In the next paragraphs, we explain a weaker (but simpler) algorithm which finds a tour of length at most twice the shortest tour. The length of the Minimum Spanning Tree of the network is a natural lower bound for the length of the optimal route. In the TSP with Triangle Inequality case it is possible to prove upper bounds in terms of the Minimum Spanning Tree and design an algorithm that has a provable upper bound on the length of the route. The first published (and the simplest) example follows.
It is easy to prove that the last step works. Moreover, thanks to the triangle inequality, each skipping at Step 4 is in fact a shortcut, i.e., the length of the cycle does not increase. Hence it gives us a TSP tour no more than twice as long as the optimal one. Christofides algorithm 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 is a long-standing (since 1975) open problem to improve 1.5 to a smaller constant. It is known, however, that there is no polynomial time algorithm that finds a tour of length at most 1/219 more than optimal, unless ''P'' = ''NP'' (Papadimitriou and Vempala, 2000). In the case of the bounded metrics, it is known, that there is no polynomial time algorithm that constructs a tour of length at most 1/388 more than optimal, unless ''P'' = ''NP'' (Engebretsen and Karpinski, 2001). The best known polynomial time approximation algorithm for the TSP problem with distances ''one'' and ''two'' finds a tour of length at most 1/7 more than optimal (Berman and Karpinski, 2006). The Christofides algorithm was one of the first Approximation Algorithm s, 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. At the time of publication, the Christofides algorithm was referred to as the Christofides heuristic. Euclidean TSP Euclidean TSP, or '''planar TSP''', is the TSP with the distance being the ordinary Euclidean Distance . Although the problem still remains NP-hard, it is known that there exists a subexponential time algorithm for it. Moreover, many heuristics work better. Euclidean TSP is a particular case of TSP with triangle inequality, since distances in plane obey triangle inequality. However, it seems to be easier than general TSP with triangle inequality. For example, the minimum spanning tree of the graph associated with an instance of 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, there is a polynomial-time algorithm that finds a tour of length at most (1 + ''c'') times the optimal for geometric instances of TSP (Arora; Mitchell; Rao and Smith); this is called a Polynomial-time Approximation Scheme . This result is an important theoretical algorithm but is not likely to be practical. Instead, heuristics with weaker guarantees are often used, but they also perform better on instances of Euclidean TSP than on general instances. Asymmetric TSP In most cases, the distance between two nodes in the TSP network is the same in both directions. The case where the distance from A to B is not equal to the distance from B to A is called asymmetric TSP. A practical application of an asymmetric TSP is route optimization using street-level routing (asymmetric due to one-way streets, slip-roads and motorways). NOTE The more common spelling in American English is ''traveling salesman problem''. REFERENCES
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