# Polynomial time solution for bipartite matching

Inspired by this StackOverflow question, I am wondering if there is an efficient algorithm for the following problem:

Assume $n$ items and $n$ boxes, with all boxes numbered numerically and all items labelled alphabetically. Each item has a disallow list: there are certain boxes that item cannot be put into. Generate a mapping of items to boxes that does not violate any item's disallow list.

For bonus points, I would be interested in being able to generate all such mappings.

At first blush, the problem appears to be the inverse of the stable marriage problem, in that everyone has a strict list (rather than an ordinal ranking) of whom they don't want to get married to but will settle for anyone else (and strictly speaking the men/boxes have no preferences). Edit: it appears this problem is closer to bipartite matching.

A very naive solution is to simply generate all permutations and check which ones don't violate any disallow lists, but this grows as $O(n!)$. Are there any polynomial time solutions out there?

This actually has nothing to do with the stable marriage problem; it's an instance of bipartite matching. (It's not related to stable marriage, becuase you don't have an ordering on the preferences of which box each item is matched to; you just have a list of what's allowed or disallowed.)

There are efficient algorithms: use the Hungarian algorithm, or any other algorithm for finding a maximum matching in a bipartite graph. See https://en.wikipedia.org/wiki/Bipartite_matching#In_unweighted_bipartite_graphs, https://en.wikipedia.org/wiki/Hungarian_algorithm, and https://en.wikipedia.org/wiki/Assignment_problem.