Science!

Standard bipartite graph construction. Binary search on the maximum possible flow which can be met (chaning capacity from source->people, and from button->sink)>

This is not necessarily tight bound? If we remove unused edges, the resulting graph of n matches is n regular. it is a "well known" result that an n-regular bipartite graph has n perfect matchings. Finding any one of them leaves us with an n-1 regular bipartite graph, which must have n-1 perfect matchings.

Proof: 1. Let G be a bipartite graph with all degrees equal to k. Show that G has a perfect matching. Solution: Fix any set X, and consider N(X). The number of edges coming out of X is exactly k|X|, but each vertex in N(X) can only absorb ≤ k many of them. So |N(X)| ≥ |X|, and we have Hall’s condition.

https://www.math.cmu.edu/~lohp/docs/math/mop2009/graph-theory-more.pdf