Restrictions on the weighted, undirected graph G=(V, E):

1. There is an edge connecting every two distinct vertices.

2. Triangular Inequality: If W(u, v) denotes the weight on the edge connecting vertex u to vertex v, then for every other vertex y,

W(u, v) W(u, y) + W(y, v).

NOTES:

• These conditions satisfy automatically by a lot of natural graph problem, e.g., cities on a planar map with weights being as-the-crow-flys (Euclidean distances).
• Even with these restrictions, the problem is still NP-hard.

Consider the following graph where the vertices happen to fall on grid points:

A simple TSP approximation algorithm algorithm:

1. Determine a Minimum Spanning Tree (MST) for G (e.g., Prim's Algorithm section 4.1)

2. Construct a path that visits every node by performing a preorder walk of the MST. (A preorder walk list a tree node every time the node is encounter including when it is first visited and "backtracked" through.)

3. Create a tour by removing vertices from the path in step 2 by taking shortcuts.

1. Determine a Minimum Spanning Tree (MST) for G (e.g., Prim's Algorithm section 4.1)

Prim's algorithm is a greedy algorithm that performs the following:

a) Select a vertex at random to be in the MST.

b) Until all the vertices are in the MST:

• Find the closest vertex not in the MST, i.e., vertex closest to any vertex in the MST
• Add this vertex using this edge to the MST

If we start with vertex 1 in the MST, determine the resulting MST.

2. Construct a path that visits every node by performing a preorder walk of the MST. (A preorder walk list a tree node every time the node is encounter including when it is first visited and "backtracked" through.)

Path constructed by a preorder walk around the MST: [ 1 2 3 8 3 2 6 5 7 5 4 5 6 2 1 ]

3. Create a tour by removing vertices from the path in step 2 by taking shortcuts.

Create the tour by taking shortcuts: [ 1 2 3 8 3 2 6 5 7 5 4 5 6 2 1 ]

Questions about the lengths:

1) If we take the optimal TSP tour and remove an edge, what do we have?

2) What is the relationship between the distance of the MST and the optimal TSP tour?

3) What is the relationship between the distance of the MST and the distance of the preorder-walk of the MST?

4) What is the relationship between the relationship between the distance of the preorder-walk of the MST and the tour obtained from the preorder-walk of the MST?

5) What is the relationship between the tour obtained from the preorder-walk the MST and the optimal TSP tour?