I have the following problem: there are 4 teams, 4 games and 4 referees. In every round, each team must be assigned to 1 game and 1 referee. After 4 rounds, each team should have seen each game and each referee, but each referee should also have seen all teams and all games.
I'm quite sure there's no solution to this problem (by trial and error), but I'm unable to see/prove why exactly. (It is possible when n=3 instead of n=4).
It seems like a 3-dimensional assignment problem without weights. The 2-dimensional assignment problem can be solved with the Hungarian algorithm; is there an equivalent to this 3-dimensional variant?
I think it could also be seen as the problem of 'filling' a 4-dimensional cube with one 1 in each column/row/etc, such that each column/row/etc has exactly 1 one.
Any insight on this type of problem (maybe it's known under a name unknown to me) would be helpful, since this is the basic version. I also have to solve the problem with other side-constraints.
Look up latin squares, in particular, you are looking for a latin hypercube (a generalization to more than two dimensions, here you have 3 -- team, game, referee).