Question
Suppose we have n cities labeled 1, 2, ..., n. For any pair of cities i and j, there is a cost C[i, j] 0
Suppose we have n cities labeled 1, 2, ..., n. For any pair of cities i and j, there is a cost C[i, j] 0 of traveling between cities i and j. Assume that these costs are symmetric and satisfy the triangle inequality. We could find a minimum spanning tree or a TSP tour in this graph - but how do the costs of those two objects relate? First, argue that the cost of a minimum cost spanning tree is at most the cost of a TSP tour (where again, a tour means that you visit every single vertex in one giant loop). Second, argue that the cost of a TSP is at most twice the cost of a minimum spanning tree (this is hard). Finally, we can find minimum spanning trees easily, but have a harder time finding TSP tours. How do the previous two parts let you bound the cost of a TSP tour easily?
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