Choosing mutually exclusive pairs efficiently

Question

This is a problem that could be done with some type of brute-force algorithm, but I was wondering if there are some efficient ways of doing this.

Let\'s assume that

Answer 1

Your question is equivalent to finding a maximum matching on a graph. The nodes of your graph are integers, and your pairs (a, b) are edges of the graph. A matching is a set of pairwise non-adjacent edges, which is equivalent to saying the same integer doesn't appear in two edges.

A polynomial time solution to this problem is the Blossom algorithm also known as Edmond's algorithm. It's rather too complicated to include the details in the answer here.