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An approximation algorithm is an algorithm that finds an approximate solution to a (typically NP-hard) problem. The quality of the algorithm is measured by how close to the actual optimum it performs. For example, it is a constant factor approximation algorithm if it always outputs a solution that is within a constant factor of the optimum. Hardness of approximation is one way to separate NP-hard problems.

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Im looking for an algorithm that can solve or approximate the solution to a problem

Your problem definition is somewhat unclear: how can "the answer to this question" not be affirmative, since for every $k$ there will be some maximum number of doors that can be opened with $k$ keys? …
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