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Combinatorial optimization typically deals with optimizing over a finite set of objects that have some combinatorial structure (e.g. trees, matchings, matroids). Approximation algorithms, polyhedral methods, and integer programming are all on topic.

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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? …
david's user avatar
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