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The nearest-neighbor algorithm is a simple heuristic used for solving the Traveling Salesman Problem (TSP). This algorithm aims to create a tour through all given points (or cities), where each point is visited exactly once before returning to the starting point. It begins by choosing an arbitrary starting point and repeatedly moving to the nearest unvisited point until all have been visited. Once this path is complete, it returns to the starting point, forming a complete loop.
While the nearest-neighbor algorithm is straightforward and fast, it's important to note that it doesn't guarantee the best possible solution, known as an optimal solution. It bases its decisions only on immediate, local information, which can sometimes result in a suboptimal overall tour. However, because of its simplicity, it is often used as a preliminary approach or when computing resources are limited.

Relative error offers a way to measure how accurate our solution is compared to the optimal solution. In the context of the TSP, we compare the cost of a tour found by an algorithm like nearest-neighbor, with the cost of the best-known tour.
In the scenario provided where the nearest-neighbor algorithm yields a tour costing \(13,500 and the optimal tour costs \)12,000, the relative error formula is applied as follows:

• Identify the Approximate Value, which is the cost found by the algorithm, here \(13,500.
• Identify the Exact Value, the cost of the optimal solution, \)12,000.
• Use the formula: \[\text{Relative Error} = \frac{\text{Approximate Value} - \text{Exact Value}}{\text{Exact Value}}\]
• Substitute the numbers: \[\frac{13,500 - 12,000}{12,000} = 0.125\]
• Convert to a percentage by multiplying by 100, resulting in a relative error of 12.5%.

This method shows the degree of deviation from the optimal solution, giving insights into the algorithm's efficiency.

In solving problems like the TSP, optimization plays a critical role. Optimization involves finding the best solution from a set of feasible ones, which for the TSP means determining the shortest tour path possible. While heuristic algorithms, such as the nearest-neighbor method, offer quick approximations, they rarely guarantee the optimized solution.
Finding an optimal solution to the TSP can be computationally intense, especially as the number of points increases. Advanced optimization techniques and algorithms are used to approach these challenges, such as Linear Programming and Metaheuristic methods like Genetic Algorithms and Simulated Annealing.
Optimization not only helps in minimizing costs or distances but also in improving the efficiency and effectiveness of solutions, which is why it's a focal point in solving complex computational problems like the TSP.