# how to generate a barcode using asp.net c# Figure 81 The optimal traveling salesman tour, shown in bold, has length 18 in Software Making Quick Response Code in Software Figure 81 The optimal traveling salesman tour, shown in bold, has length 18

Figure 81 The optimal traveling salesman tour, shown in bold, has length 18
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components of a particular graph constructed from the instance (recall Exercise 328) In fact, in 9, we ll see a different polynomial algorithm for this same special case, which is called 2 SAT On the other hand, if we are just a little more permissive and allow clauses to contain three literals, then the resulting problem, known as 3 SAT (an example of which we saw earlier), once again becomes hard to solve!
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The traveling salesman problem
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In the traveling salesman problem (TSP) we are given n vertices 1, , n and all n(n 1)/2 distances between them, as well as a budget b We are asked to nd a tour, a cycle that passes through every vertex exactly once, of total cost b or less or to report that no such tour exists That is, we seek a permutation (1), , (n) of the vertices such that when they are toured in this order, the total distance covered is at most b: d (1), (2) + d (2), (3) + + d (n), (1) b
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See Figure 81 for an example (only some of the distances are shown; assume the rest are very large) Notice how we have de ned the TSP as a search problem: given an instance, nd a tour within the budget (or report that none exists) But why are we expressing the traveling salesman problem in this way, when in reality it is an optimization problem, in which the shortest possible tour is sought Why dress it up as something else For a good reason Our plan in this chapter is to compare and relate problems The framework of search problems is helpful in this regard, because it encompasses optimization problems like the TSP in addition to true search problems like SAT Turning an optimization problem into a search problem does not change its dif culty at all, because the two versions reduce to one another Any algorithm that solves the optimization TSP also readily solves the search problem: nd the optimum tour and if it is within budget, return it; if not, there is no solution Conversely, an algorithm for the search problem can also be used to solve the optimization problem To see why, rst suppose that we somehow knew the cost of the optimum tour; then we could nd this tour by calling the algorithm for the search problem, using the optimum 232
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cost as the budget Fine, but how do we nd the optimum cost Easy: By binary search! (See Exercise 81) Incidentally, there is a subtlety here: Why do we have to introduce a budget Isn t any optimization problem also a search problem in the sense that we are searching for a solution that has the property of being optimal The catch is that the solution to a search problem should be easy to recognize, or as we put it earlier, polynomial-time checkable Given a potential solution to the TSP, it is easy to check the properties is a tour (just check that each vertex is visited exactly once) and has total length b But how could one check the property is optimal As with SAT, there are no known polynomial-time algorithms for the TSP, despite much effort by researchers over nearly a century Of course, there is an exponential algorithm for solving it, by trying all (n 1)! tours, and in Section 66 we saw a faster, yet still exponential, dynamic programming algorithm The minimum spanning tree ( MST ) problem, for which we do have ef cient algorithms, provides a stark contrast here To phrase it as a search problem, we are again given a distance matrix and a bound b, and are asked to nd a tree T with total weight (i,j) T dij b The TSP can be thought of as a tough cousin of the MST problem, in which the tree is not allowed to branch and is therefore a path1 This extra restriction on the structure of the tree results in a much harder problem
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