How to check for a unique distribution in a matrix?

Question

Algorithms

The third day I struggle with the decision:
It is necessary to understand whether there is such a distribution of workers with a set of skills for work, so that all work is completed.
He reduced everything to a task:
there are n people and m skills.
We believe that each person can have from 1 to m skills
. We need to understand if there is at least one such distribution of people that all skills will be covered.
In fact, an m by n matrix of ones in random places.
We assume that there are more people than or equal to the number of jobs, otherwise it is obvious that there is no such distribution.
The simplest case:
1 1 1
1 0 0
1 0 0
three people, one can do everything, others can do the same. - They are not enough
I supposed to look for the rank of the matrix so that it was not less than the number of skills, but the rank of the matrix
111
111
111
is equal to one, and such a set of workers corresponds to the distribution conditions.

Answer 1

The Hungarian algorithm solves a more general problem.