Apply to Figure 1a the following approximation algorithms (Johnson and Papadimitriou 1985;...

Apply to Figure 1a the following approximation algorithms (Johnson and Papadimitriou 1985; Rosenkrantz, Stearns, and Lewis, 1977) to solve the traveling salesman problem.

a. The nearest neighbor algorithm (next best method) begins with an arbitrary vertex v and then finds a vertex w not on the tour that is closest to the vertex u last added and includes in the tour edge(uw) and edge(wv) after deleting edge(uv).

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b. The nearest insertion algorithm is obtained from nearestAdditionAlgorithm() by finding two vertices Vq and Vr in the tour that minimize the expression

dist(edge(vqvi)) + dist(edge(vivr)) – dist(edge(vqvr))

In this way, a new vertex vi is inserted in the best place in the existing tour, which may not be next to vp.

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c. The cheapest insertion algorithm is obtained from nearestAdditionAlgorithm() by including in tour a new vertex vi that minimizes the length of the new tour.

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d. The farthest insertion algorithm is just like the nearest insertion algorithms except that it requires that vi is farthest from tour, not closest.

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e. The nearest merger algorithm, which corresponds to the Borůvka algorithm:

Figure 1a Using a minimum spanning tree to find a minimum salesman tour.