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MIT6_042JF10_rec09

MIT6_042JF10_rec09 - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science October 8, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 9 1 Traveling Salesperson Problem Now we’re going to talk about a famous optimization problem known as the Traveling Sales- person Problem 1 (TSP). 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 returns to the starting city? One special, though very natural, case of this problem is when the costs of traveling between cities obey the triangle inequality . That is, if a, b, and c are distinct cities, then d ( a, c ) d ( a, b ) + d ( b, c ). This corresponds to our intuitive notion that the distance of traveling to a from c should be no larger than that of first traveling to a from b , and then from b to c . The triangle inequality holds if the cities correspond to points in the plane, since in this case the line segments joining a to b , b to c , and a to c form a triangle, and in high school we learned that the sum of any two edges of a triangle is larger than the third edge. Let us abstract away the distractions and formulate the problem as follows. The cities will be vertices of a graph. We will then consider the complete graph on these vertices, and give edge ( a, b ) a weight of d ( a, b ). We want an ordering of the vertices ( v 1 , . . . , v n ) that minimizes n 1 COST = d ( v i , v i +1 ) + d ( v n , v 1 ) .
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