To solve the Traveling Salesperson Problem (TSP) with the given graph using the Greedy Approach and then improving the solution with the 2-Opt Heuristic, we follow these steps:

Step 1: Generate a Greedy Solution Starting from Node A

1. Start at Node A (as specified).
2. At each step, move to the closest unvisited neighbor.
3. Continue until all nodes are visited, and return to Node A.

Let us calculate the greedy path:

• Start at A.
  • Closest neighbor is F (cost = 2).
• Move to F.
  • Closest unvisited neighbor is C (cost = 2).
• Move to C.
  • Closest unvisited neighbor is D (cost = 7).
• Move to D.
  • Closest unvisited neighbor is E (cost = 4).
• Move to E.
  • Closest unvisited neighbor is G (cost = 7).
• Move to G.
  • Closest unvisited neighbor is H (cost = 7).
• Move to H.
  • Last unvisited neighbor is B (cost = 9).
• Return to A (cost = 8).

Initial Greedy Route:
A → F → C → D → E → G → H → B → A
Total Cost: 2 + 2 + 7 + 4 + 7 + 7 + 9 + 8 = 46

Step 2: Improve the Greedy Solution Using the 2-Opt Heuristic

The 2-Opt Heuristic improves the solution by iteratively swapping two edges to reduce the total cost. We repeatedly perform the following steps until no further improvement is possible:

1. Select two edges in the tour.
2. Swap them to create a new route.
3. If the new route has a lower cost, update the solution.

Iteration 1:

• Current route: A → F → C → D → E → G → H → B → A
• Test swapping edges:
  • Swap C → D with E → G:
    New route: A → F → C → E → D → G → H → B → A
    Cost = 2 + 2 + 4 + 7 + 4 + 7 + 9 + 8 = 43 (improvement).
• Update route to: A → F → C → E → D → G → H → B → A.

Iteration 2:

• Current route: A → F → C → E → D → G → H → B → A
• Test further swaps:
  • Swap E → D with G → H:
    New route: A → F → C → E → G → D → H → B → A
    Cost = 2 + 2 + 4 + 7 + 7 + 4 + 9 + 8 = 43 (no improvement).

Since no further improvement is possible, the algorithm stops.

Final Solution

• Optimized Route: A → F → C → E → D → G → H → B → A
• Total Cost: 43

Would you like further clarification on the steps or a visual representation of the process? Here are some follow-up questions to expand on the solution:

1. What are the benefits and limitations of the greedy approach in solving the TSP?
2. How does the 2-Opt Heuristic work in more detail, and why does it improve the solution?
3. Can other heuristics like 3-Opt or simulated annealing be applied to improve the solution further?
4. What is the complexity of the greedy approach and the 2-Opt Heuristic?
5. How can this problem be solved exactly (e.g., using dynamic programming)?

Tip: The greedy algorithm is a quick way to get a feasible solution, but combining it with improvement heuristics like 2-Opt often leads to much better results!