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Lec29_30_TravelingSalesman_Approximation_Parallel

# Lec29_30_TravelingSalesman_Approximation_Parallel -...

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Traveling Salesman Problem

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Problem statement A salesman spends his time visiting n cities (or nodes) cyclically. In one tour he visits each city just once, and finishes up where he started. In what order should he visit them to minimize the distance traveled? Find an optimal tour in a weighted, directed graph so that the path has minimum length A tour in a directed graph is a path from a vertex to itself that passes through each of other vertices exactly once
TSP If there are only 2 cities then the problem is trivial, since only one tour is possible. For the n cities, if all links are present then there are (n-1)! different tours. To see why this is so, pick any city as the first - then there are n-1 choices for the second city visited, n-2 choices for the third, and so on. The number of solutions becomes extremely large for large n, so that an exhaustive search is impractical.

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Importance of TSP The problem has some direct importance, since quite a lot of practical applications can be put in this form. It also has a theoretical importance in complexity theory, since the TSP is one of the class of "NP Complete" combinatorial problems. NP Complete problems have intractable in the sense that no one has found any really efficient way of solving them for large n. They are also known to be more or less equivalent to each other; if you knew how to solve one kind of NP Complete problem you could solve the lot.
Optimal? Exponential time complexity Can we find a solution algorithm that gives an optimal solution in a time that has a polynomial variation with the size n of the problem.

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