What does "Approximation Ratio" mean?

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The approximation ratio is a way to measure how good a solution is compared to the best possible answer for a problem. When solving complex problems, especially in areas like computer science and mathematics, finding the exact solution can be very hard or take too much time.

Instead, we often look for solutions that are "close enough" to the best answer. The approximation ratio gives us a number that tells us just how close our solution is. A lower ratio means our solution is better when compared to the best possible solution.

For example, if we have a problem and our solution costs us 10 units, while the best solution would cost 5 units, our approximation ratio would be 2. This means our solution is twice as costly as the best one.

In some cases, improvements in algorithms can lead to better approximation ratios. This means that the new methods can find solutions that are closer to the best answer than before, making them more efficient and effective in solving problems.