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Some of the most well-known problems within Computer Science are virtually impossible to solve optimally at scale in a useful time. One of the most important known problems is the Traveling Salesmen problem. This problem boils down to: given a list of cities and the distances between each pair of them, what’s the shortest possible route to go from a certain city to itself, visiting all of the other cities. This is problem has many more applications that its literal sense, hence its importance.
Those kind of problems are called NP-hard problems and there is no algorithm that solves them in polynomial time, which leaves us with exponential algorithms or worse. For small datasets, that might not be a problem. However, to solve a big problem, with a large dataset we can’t afford to use slow algorithms.
Thus, there exist two ways of solving this problems even though they’re not optimal:
- Use an heuristic, usually by some observation such as “I noticed that this happened, so let’s try for the rest”, even though there’s no proof of it. It may or may not give a good solution, but there’s no guarantee whatsoever.
- Use an Approximation Algorithms, which guarantees that the solution is within a certain range from the optimal solution and that can be proved.
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